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3: TWO-DIMENSIONAL KINEMATICS Boundless (now LumenLearning) Boundless CHAPTER OVERVIEW 3: Two-Dimensional Kinematics 3.1: Motion in Two Dimensions 3.2: Vectors 3.3: Projectile Motion 3.4: Multiple Velocities This page titled 3: Two-Dimensional Kinematics is shared under a not declared license an...

3: TWO-DIMENSIONAL KINEMATICS Boundless (now LumenLearning) Boundless CHAPTER OVERVIEW 3: Two-Dimensional Kinematics 3.1: Motion in Two Dimensions 3.2: Vectors 3.3: Projectile Motion 3.4: Multiple Velocities This page titled 3: Two-Dimensional Kinematics is shared under a not declared license and was authored, remixed, and/or curated by Boundless. 1 3.1: Motion in Two Dimensions Constant Velocity An object moving with constant velocity must have a constant speed in a constant direction. learning objectives Examine the terms for constant velocity and how they apply to acceleration Motion with constant velocity is one of the simplest forms of motion. This type of motion occurs when an an object is moving (or sliding) in the presence of little or negligible friction, similar to that of a hockey puck sliding across the ice. To have a constant velocity, an object must have a constant speed in a constant direction. Constant direction constrains the object to motion to a straight path. Newton’s second law (F = ma ) suggests that when a force is applied to an object, the object would experience acceleration. If the acceleration is 0, the object shouldn’t have any external forces applied on it. Mathematically, this can be shown as the following: dv a = = 0 ⇒ v = const. (3.1.1) dt If an object is moving at constant velocity, the graph of distance vs. time (x vs. t ) shows the same change in position over each interval of time. Therefore the motion of an object at constant velocity is represented by a straight line: x = x + vt , where x is the displacement when t = 0 (or at the y-axis intercept). 0 0 Motion with Constant Velocity: When an object is moving with constant velocity, it does not change direction nor speed and therefore is represented as a straight line when graphed as distance over time. You can also obtain an object’s velocity if you know its trace over time. Given a graph as in, we can calculate the velocity from the change in distance over the change in time. In graphical terms, the velocity can be interpreted as the slope of the line. The velocity can be positive or negative, and is indicated by the sign of our slope. This tells us in which direction the object moves. Constant Acceleration Analyzing two-dimensional projectile motion is done by breaking it into two motions: along the horizontal and vertical axes. learning objectives Analyze a two-dimensional projectile motion along horizontal and vertical axes Projectile motion is the motion of an object thrown, or projected, into the air, subject only to the force of gravity. The object is called a projectile, and its path is called its trajectory. The motion of falling objects is a simple one-dimensional type of projectile motion in which there is no horizontal movement. In two-dimensional projectile motion, such as that of a football or other thrown object, there is both a vertical and a horizontal component to the motion. 3.1.1 https://phys.libretexts.org/@go/page/14445 Projectile Motion: Throwing a rock or kicking a ball generally produces a projectile pattern of motion that has both a vertical and a horizontal component. The most important fact to remember is that motion along perpendicular axes are independent and thus can be analyzed separately. The key to analyzing two-dimensional projectile motion is to break it into two motions, one along the horizontal axis and the other along the vertical. To describe motion we must deal with velocity and acceleration, as well as with displacement. We will assume all forces except for gravity (such as air resistance and friction, for example) are negligible. The components of acceleration are then very simple: a = −g = −9.81 (we assume that the motion occurs at small enough heights near the surface of the earth so that the acceleration due to gravity is constant). Because the acceleration due to gravity is along the vertical direction only, a = 0 . Thus, the kinematic equations describing the motion along the x and y directions respectively, can be used: m y 2 s x x = x0 + vx t (3.1.2) y = v0y + ay t (3.1.3) 1 y = y0 + v0y t + 2 vy 2 =v 0y 2 2 ay t (3.1.4) + 2 ay (y − y0 ) (3.1.5) We analyze two-dimensional projectile motion by breaking it into two independent one-dimensional motions along the vertical and horizontal axes. The horizontal motion is simple, because a = 0 and v is thus constant. The velocity in the vertical direction begins to decrease as an object rises; at its highest point, the vertical velocity is zero. As an object falls towards the Earth again, the vertical velocity increases again in magnitude but points in the opposite direction to the initial vertical velocity. The xx and yy motions can be recombined to give the total velocity at any given point on the trajectory. x x Key Points Constant velocity means that the object in motion is moving in a straight line at a constant speed. This line can be represented algebraically as: x = x + vt , where x represents the position of the object at t = 0 , and the slope of the line indicates the object’s speed. The velocity can be positive or negative, and is indicated by the sign of our slope. This tells us in which direction the object moves. Constant acceleration in motion in two dimensions generally follows a projectile pattern. Projectile motion is the motion of an object thrown or projected into the air, subject to only the (vertical) acceleration due to gravity. We analyze two-dimensional projectile motion by breaking it into two independent one-dimensional motions along the vertical and horizontal axes. 0 0 Key Terms constant velocity: Motion that does not change in speed nor direction. kinematic: of or relating to motion or kinematics LICENSES AND ATTRIBUTIONS CC LICENSED CONTENT, SHARED PREVIOUSLY Curation and Revision. Provided by: Boundless.com. License: CC BY-SA: Attribution-ShareAlike 3.1.2 https://phys.libretexts.org/@go/page/14445 CC LICENSED CONTENT, SPECIFIC ATTRIBUTION Provided by: Light and Matter. Located at: http://lightandmatter.com/lmb.pdf. License: CC BY: Attribution Velocity. Provided by: Wikipedia. Located at: http://en.Wikipedia.org/wiki/Velocity. License: CC BY-SA: AttributionShareAlike Boundless. Provided by: Boundless Learning. Located at: www.boundless.com//physics/definition/constant-velocity. License: CC BY-SA: Attribution-ShareAlike Provided by: Light and Matter. Located at: http://lightandmatter.com/lmb.pdf. License: CC BY: Attribution Acceleration. Provided by: Wikipedia. Located at: http://en.Wikipedia.org/wiki/Acceleration. License: CC BY-SA: Attribution-ShareAlike OpenStax College, College Physics. September 17, 2013. Provided by: OpenStax CNX. Located at: http://cnx.org/content/m42042/latest/?collection=col11406/1.7. License: CC BY: Attribution kinematic. Provided by: Wiktionary. Located at: http://en.wiktionary.org/wiki/kinematic. License: CC BY-SA: AttributionShareAlike Provided by: Light and Matter. Located at: http://lightandmatter.com/lmb.pdf. License: CC BY: Attribution OpenStax College, College Physics. October 19, 2012. Provided by: OpenStax CNX. Located at: http://cnx.org/content/m42042/latest/?collection=col11406/1.7. License: CC BY: Attribution This page titled 3.1: Motion in Two Dimensions is shared under a not declared license and was authored, remixed, and/or curated by Boundless. 3.1.3 https://phys.libretexts.org/@go/page/14445 3.2: Vectors Components of a Vector Vectors are geometric representations of magnitude and direction and can be expressed as arrows in two or three dimensions. learning objectives Contrast two-dimensional and three-dimensional vectors Vectors are geometric representations of magnitude and direction which are often represented by straight arrows, starting at one point on a coordinate axis and ending at a different point. All vectors have a length, called the magnitude, which represents some quality of interest so that the vector may be compared to another vector. Vectors, being arrows, also have a direction. This differentiates them from scalars, which are mere numbers without a direction. A vector is defined by its magnitude and its orientation with respect to a set of coordinates. It is often useful in analyzing vectors to break them into their component parts. For two-dimensional vectors, these components are horizontal and vertical. For three dimensional vectors, the magnitude component is the same, but the direction component is expressed in terms of xx, yy and zz. Decomposing a Vector To visualize the process of decomposing a vector into its components, begin by drawing the vector from the origin of a set of coordinates. Next, draw a straight line from the origin along the x-axis until the line is even with the tip of the original vector. This is the horizontal component of the vector. To find the vertical component, draw a line straight up from the end of the horizontal vector until you reach the tip of the original vector. You should find you have a right triangle such that the original vector is the hypotenuse. Decomposing a vector into horizontal and vertical components is a very useful technique in understanding physics problems. Whenever you see motion at an angle, you should think of it as moving horizontally and vertically at the same time. Simplifying vectors in this way can speed calculations and help to keep track of the motion of objects. Scalars and Vectors 3.2.1 https://phys.libretexts.org/@go/page/14446 Scalars and Vectors: Mr. Andersen explains the differences between scalar and vectors quantities. He also uses a demonstration to show the importance of vectors and vector addition. Components of a Vector: The original vector, defined relative to a set of axes. The horizontal component stretches from the start of the vector to its furthest x-coordinate. The vertical component stretches from the x-axis to the most vertical point on the vector. Together, the two components and the vector form a right triangle. Scalars vs. Vectors Scalars are physical quantities represented by a single number, and vectors are represented by both a number and a direction. learning objectives Distinguish the difference between the quantities scalars and vectors represent Physical quantities can usually be placed into two categories, vectors and scalars. These two categories are typified by what information they require. Vectors require two pieces of information: the magnitude and direction. In contrast, scalars require only the magnitude. Scalars can be thought of as numbers, whereas vectors must be thought of more like arrows pointing in a specific direction. 3.2.2 https://phys.libretexts.org/@go/page/14446 A Vector: An example of a vector. Vectors are usually represented by arrows with their length representing the magnitude and their direction represented by the direction the arrow points. Vectors require both a magnitude and a direction. The magnitude of a vector is a number for comparing one vector to another. In the geometric interpretation of a vector the vector is represented by an arrow. The arrow has two parts that define it. The two parts are its length which represents the magnitude and its direction with respect to some set of coordinate axes. The greater the magnitude, the longer the arrow. Physical concepts such as displacement, velocity, and acceleration are all examples of quantities that can be represented by vectors. Each of these quantities has both a magnitude (how far or how fast) and a direction. In order to specify a direction, there must be something to which the direction is relative. Typically this reference point is a set of coordinate axes like the x-y plane. Scalars differ from vectors in that they do not have a direction. Scalars are used primarily to represent physical quantities for which a direction does not make sense. Some examples of these are: mass, height, length, volume, and area. Talking about the direction of these quantities has no meaning and so they cannot be expressed as vectors. The difference between Vectors and Scalars, Introd… Introd… 3.2.3 https://phys.libretexts.org/@go/page/14446 The difference between Vectors and Scalars, Introduction and Basics: This video introduces the difference between scalars and vectors. Ideas about magnitude and direction are introduced and examples of both vectors and scalars are given. Adding and Subtracting Vectors Graphically Vectors may be added or subtracted graphically by laying them end to end on a set of axes. learning objectives Distinguish the difference between the quantities scalars and vectors represent Adding and Subtracting Vectors One of the ways in which representing physical quantities as vectors makes analysis easier is the ease with which vectors may be added to one another. Since vectors are graphical visualizations, addition and subtraction of vectors can be done graphically. The graphical method of vector addition is also known as the head-to-tail method. To start, draw a set of coordinate axes. Next, draw out the first vector with its tail (base) at the origin of the coordinate axes. For vector addition it does not matter which vector you draw first since addition is commutative, but for subtraction ensure that the vector you draw first is the one you are subtracting from. The next step is to take the next vector and draw it such that its tail starts at the previous vector’s head (the arrow side). Continue to place each vector at the head of the preceding one until all the vectors you wish to add are joined together. Finally, draw a straight line from the origin to the head of the final vector in the chain. This new line is the vector result of adding those vectors together. Graphical Addition of Vectors: The head-to-tail method of vector addition requires that you lay out the first vector along a set of coordinate axes. Next, place the tail of the next vector on the head of the first one. Draw a new vector from the origin to the head of the last vector. This new vector is the sum of the original two. 3.2.4 https://phys.libretexts.org/@go/page/14446 Vector Addition Lesson 1 of 2: Head to Tail Addition … Vector Addition Lesson 1 of 2: Head to Tail Addition Method: This video gets viewers started with vector addition and subtraction. The first lesson shows graphical addition while the second video takes a more mathematical approach and shows vector addition by components. To subtract vectors the method is similar. Make sure that the first vector you draw is the one to be subtracted from. Then, to subtract a vector, proceed as if adding the opposite of that vector. In other words, flip the vector to be subtracted across the axes and then join it tail to head as if adding. To flip the vector, simply put its head where its tail was and its tail where its head was. Adding and Subtracting Vectors Using Components It is often simpler to add or subtract vectors by using their components. learning objectives Demonstrate how to add and subtract vectors by components Using Components to Add and Subtract Vectors Another way of adding vectors is to add the components. Previously, we saw that vectors can be expressed in terms of their horizontal and vertical components. To add vectors, merely express both of them in terms of their horizontal and vertical 3.2.5 https://phys.libretexts.org/@go/page/14446 components and then add the components together. Vector with Horizontal and Vertical Components: The vector in this image has a magnitude of 10.3 units and a direction of 29.1 degrees above the x-axis. It can be decomposed into a horizontal part and a vertical part as shown. For example, a vector with a length of 5 at a 36.9 degree angle to the horizontal axis will have a horizontal component of 4 units and a vertical component of 3 units. If we were to add this to another vector of the same magnitude and direction, we would get a vector twice as long at the same angle. This can be seen by adding the horizontal components of the two vectors (4 + 4) and the two vertical components (3 + 3 ). These additions give a new vector with a horizontal component of 8(4 + 4) and a vertical component of 6(3 + 3) . To find the resultant vector, simply place the tail of the vertical component at the head (arrow side) of the horizontal component and then draw a line from the origin to the head of the vertical component. This new line is the resultant vector. It should be twice as long as the original, since both of its components are twice as large as they were previously. To subtract vectors by components, simply subtract the two horizontal components from each other and do the same for the vertical components. Then draw the resultant vector as you did in the previous part. Vector Addition Lesson 2 of 2: How to Add Vectors b… b… 3.2.6 https://phys.libretexts.org/@go/page/14446 Vector Addition Lesson 2 of 2: How to Add Vectors by Components: This video gets viewers started with vector addition using a mathematical approach and shows vector addition by components. Multiplying Vectors by a Scalar Multiplying a vector by a scalar changes the magnitude of the vector but not the direction. learning objectives Summarize the interaction between vectors and scalars Overview Although vectors and scalars represent different types of physical quantities, it is sometimes necessary for them to interact. While adding a scalar to a vector is impossible because of their different dimensions in space, it is possible to multiply a vector by a scalar. A scalar, however, cannot be multiplied by a vector. To multiply a vector by a scalar, simply multiply the similar components, that is, the vector’s magnitude by the scalar’s magnitude. This will result in a new vector with the same direction but the product of the two magnitudes. Example 3.2.1: For example, if you have a vector A with a certain magnitude and direction, multiplying it by a scalar a with magnitude 0.5 will give a new vector with a magnitude of half the original. Similarly if you take the number 3 which is a pure and unit-less scalar and multiply it to a vector, you get a version of the original vector which is 3 times as long. As a more physical example take the gravitational force on an object. The force is a vector with its magnitude depending on the scalar known as mass and its direction being down. If the mass of the object is doubled, the force of gravity is doubled as well. Multiplying vectors by scalars is very useful in physics. Most of the units used in vector quantities are intrinsically scalars multiplied by the vector. For example, the unit of meters per second used in velocity, which is a vector, is made up of two scalars, which are magnitudes: the scalar of length in meters and the scalar of time in seconds. In order to make this conversion from magnitudes to velocity, one must multiply the unit vector in a particular direction by these scalars. 3.2.7 https://phys.libretexts.org/@go/page/14446 Scalar Multiplication: (i) Multiplying the vector A by the scalar a = 0.5 yields the vector B which is half as long. (ii) Multiplying the vector A by 3 triples its length. (iii) Doubling the mass (scalar) doubles the force (vector) of gravity. Unit Vectors and Multiplication by a Scalar Multiplying a vector by a scalar is the same as multiplying its magnitude by a number. learning objectives Predict the influence of multiplying a vector by a scalar In addition to adding vectors, vectors can also be multiplied by constants known as scalars. Scalars are distinct from vectors in that they are represented by a magnitude but no direction. Examples of scalars include an object’s mass, height, or volume. Scalar Multiplication: (i) Multiplying the vector A by the scalar a = 0.5 yields the vector B which is half as long. (ii) Multiplying the vector A by 3 triples its length. (iii) Doubling the mass (scalar) doubles the force (vector) of gravity. When multiplying a vector by a scalar, the direction of the vector is unchanged and the magnitude is multiplied by the magnitude of the scalar. This results in a new vector arrow pointing in the same direction as the old one but with a longer or shorter length. You can also accomplish scalar multiplication through the use of a vector’s components. Once you have the vector’s components, multiply each of the components by the scalar to get the new components and thus the new vector. A useful concept in the study of vectors and geometry is the concept of a unit vector. A unit vector is a vector with a length or magnitude of one. The unit vectors are different for different coordinates. In Cartesian coordinates the directions are x and y ^ and y ^ . With the triangle above the letters referred to as a “hat”. The unit vectors in Cartesian coordinates usually denoted x describe a circle known as the “unit circle” which has radius one. This can be seen by taking all the possible vectors of length one at all the possible angles in this coordinate system and placing them on the coordinates. If you were to draw a line around connecting all the heads of all the vectors together, you would get a circle of radius one. Position, Displacement, Velocity, and Acceleration as Vectors Position, displacement, velocity, and acceleration can all be shown vectors since they are defined in terms of a magnitude and a direction. learning objectives Examine the applications of vectors in analyzing physical quantities Use of Vectors Vectors can be used to represent physical quantities. Most commonly in physics, vectors are used to represent displacement, velocity, and acceleration. Vectors are a combination of magnitude and direction, and are drawn as arrows. The length represents the magnitude and the direction of that quantity is the direction in which the vector is pointing. Because vectors are constructed this way, it is helpful to analyze physical quantities (with both size and direction) as vectors. Applications In physics, vectors are useful because they can visually represent position, displacement, velocity and acceleration. When drawing vectors, you often do not have enough space to draw them to the scale they are representing, so it is important to denote somewhere 3.2.8 https://phys.libretexts.org/@go/page/14446 what scale they are being drawn at. For example, when drawing a vector that represents a magnitude of 100, one may draw a line that is 5 units long at a scale of . When the inverse of the scale is multiplied by the drawn magnitude, it should equal the actual magnitude. 1 20 Position and Displacement Displacement is defined as the distance, in any direction, of an object relative to the position of another object. Physicists use the concept of a position vector as a graphical tool to visualize displacements. A position vector expresses the position of an object from the origin of a coordinate system. A position vector can also be used to show the position of an object in relation to a reference point, secondary object or initial position (if analyzing how far the object has moved from its original location). The position vector is a straight line drawn from the arbitrary origin to the object. Once drawn, the vector has a length and a direction relative to the coordinate system used. Velocity Velocity is also defined in terms of a magnitude and direction. To say that something is gaining or losing velocity one must also say how much and in what direction. For example, an airplane flying at 200 to the northeast can be represented by an vector pointing in the northeast direction with a magnitude of 200 . In drawing the vector, the magnitude is only important as a way to compare two vectors of the same units. So, if there were another airplane flying 100 to the southwest, the vector arrow should be half as long and pointing in the direction of southwest. km h km h km h Acceleration Acceleration, being the time rate of change of velocity, is composed of a magnitude and a direction, and is drawn with the same concept as a velocity vector. A value for acceleration would not be helpful in physics if the magnitude and direction of this acceleration was unknown, which is why these vectors are important. In a free body diagram, for example, of an object falling, it would be helpful to use an acceleration vector near the object to denote its acceleration towards the ground. If gravity is the only force acting on the object, this vector would be pointing downward with a magnitude of 9.81 of 32.2 . m ft s2 s2 Vector Diagram: Here is a man walking up a hill. His direction of travel is defined by the angle theta relative to the vertical axis and by the length of the arrow going up the hill. He is also being accelerated downward by gravity. Key Points Vectors can be broken down into two components: magnitude and direction. By taking the vector to be analyzed as the hypotenuse, the horizontal and vertical components can be found by completing a right triangle. The bottom edge of the triangle is the horizontal component and the side opposite the angle is the vertical component. The angle that the vector makes with the horizontal can be used to calculate the length of the two components. Scalars are physical quantities represented by a single number and no direction. Vectors are physical quantities that require both magnitude and direction. Examples of scalars include height, mass, area, and volume. Examples of vectors include displacement, velocity, and acceleration. To add vectors, lay the first one on a set of axes with its tail at the origin. Place the next vector with its tail at the previous vector’s head. When there are no more vectors, draw a straight line from the origin to the head of the last vector. This line is the sum of the vectors. To subtract vectors, proceed as if adding the two vectors, but flip the vector to be subtracted across the axes and then join it tail to head as if adding. 3.2.9 https://phys.libretexts.org/@go/page/14446 Adding or subtracting any number of vectors yields a resultant vector. Vectors can be decomposed into horizontal and vertical components. Once the vectors are decomposed into components, the components can be added. Adding the respective components of two vectors yields a vector which is the sum of the two vectors. A vector is a quantity with both magnitude and direction. A scalar is a quantity with only magnitude. Multiplying a vector by a scalar is equivalent to multiplying the vector’s magnitude by the scalar. The vector lengthens or shrinks but does not change direction. A unit vector is a vector of magnitude ( length ) 1. A scalar is a physical quantity that can be represented by a single number. Unlike vectors, scalars do not have direction. Multiplying a vector by a scalar is the same as multiplying the vector’s magnitude by the number represented by the scalar. Vectors are arrows consisting of a magnitude and a direction. They are used in physics to represent physical quantities that also have both magnitude and direction. Displacement is a physics term meaning the distance of an object from a reference point. Since the displacement contains two pieces of information: the distance from the reference point and the direction away from the point, it is well represented by a vector. Velocity is defined as the rate of change in time of the displacement. To know the velocity of an object one must know both how fast the displacement is changing and in what direction. Therefore it is also well represented by a vector. Acceleration, being the rate of change of velocity also requires both a magnitude and a direction relative to some coordinates. When drawing vectors, you often do not have enough space to draw them to the scale they are representing, so it is important to denote somewhere what scale they are being drawn at. Key Terms coordinates: Numbers indicating a position with respect to some axis. Ex: x and y coordinates indicate position relative to xx and yy axes. axis: An imaginary line around which an object spins or is symmetrically arranged. magnitude: A number assigned to a vector indicating its length. Coordinate axes: A set of perpendicular lines which define coordinates relative to an origin. Example: x and y coordinate axes define horizontal and vertical position. origin: The center of a coordinate axis, defined as being the coordinate 0 in all axes. Component: A part of a vector. For example, horizontal and vertical components. vector: A directed quantity, one with both magnitude and direction; the between two points. magnitude: A number assigned to a vector indicating its length. scalar: A quantity that has magnitude but not direction; compare vector. unit vector: A vector of magnitude 1. velocity: The rate of change of displacement with respect to change in time. displacement: The length and direction of a straight line between two objects. acceleration: the rate at which the velocity of a body changes with time LICENSES AND ATTRIBUTIONS CC LICENSED CONTENT, SHARED PREVIOUSLY Curation and Revision. Provided by: Boundless.com. License: CC BY-SA: Attribution-ShareAlike CC LICENSED CONTENT, SPECIFIC ATTRIBUTION Sunil Kumar Singh, Components of a Vector. September 17, 2013. Provided by: OpenStax CNX. Located at: http://cnx.org/content/m14519/latest/. License: CC BY: Attribution Euclidean vector. Provided by: Wikipedia. Located at: en.Wikipedia.org/wiki/Euclidean_vector. License: CC BY-SA: Attribution-ShareAlike axis. Provided by: Wiktionary. Located at: en.wiktionary.org/wiki/axis. License: CC BY-SA: Attribution-ShareAlike coordinates. Provided by: Wikipedia. Located at: en.Wikipedia.org/wiki/coordinates. License: CC BY-SA: AttributionShareAlike 3.2.10 https://phys.libretexts.org/@go/page/14446 magnitude. Provided by: Wiktionary. Located at: en.wiktionary.org/wiki/magnitude. License: CC BY-SA: AttributionShareAlike Scalars and Vectors. Located at: http://www.youtube.com/watch?v=EUrMI0DIh40. License: Public Domain: No Known Copyright. License Terms: Standard YouTube license Boundless. Provided by: Amazon Web Services. Located at: s3.amazonaws.com/figures.boundless.com/5101a2b3e4b04253d8aba44b/vectordecomp.jpg. License: CC BY: Attribution Scalar (physics). Provided by: Wikipedia. Located at: en.Wikipedia.org/wiki/Scalar_(physics). License: CC BY-SA: Attribution-ShareAlike Euclidean vector. Provided by: Wikipedia. Located at: en.Wikipedia.org/wiki/Euclidean_vector. License: CC BY-SA: Attribution-ShareAlike Boundless. Provided by: Boundless Learning. Located at: www.boundless.com//physics/definition/coordinate-axes. License: CC BY-SA: Attribution-ShareAlike Scalars and Vectors. Located at: http://www.youtube.com/watch?v=EUrMI0DIh40. License: Public Domain: No Known Copyright. License Terms: Standard YouTube license Boundless. Provided by: Amazon Web Services. Located at: s3.amazonaws.com/figures.boundless.com/5101a2b3e4b04253d8aba44b/vectordecomp.jpg. License: CC BY: Attribution Provided by: Wikimedia. Located at: upload.wikimedia.org/Wikipedia/commons/thumb/5/5d/Position_vector.svg/220pxPosition_vector.svg.png. License: CC BY: Attribution The difference between Vectors and Scalars, Introduction and Basics. Located at: http://www.youtube.com/watch? v=bap6XjDDE3k. License: Public Domain: No Known Copyright. License Terms: Standard YouTube license OpenStax College, Vector Addition and Subtraction: Graphical Methods. September 17, 2013. Provided by: OpenStax CNX. Located at: http://cnx.org/content/m42127/latest/. License: CC BY: Attribution Boundless. Provided by: Boundless Learning. Located at: www.boundless.com//physics/definition/coordinate-axes. License: CC BY-SA: Attribution-ShareAlike origin. Provided by: Wiktionary. Located at: en.wiktionary.org/wiki/origin. License: CC BY-SA: Attribution-ShareAlike Scalars and Vectors. Located at: http://www.youtube.com/watch?v=EUrMI0DIh40. License: Public Domain: No Known Copyright. License Terms: Standard YouTube license Boundless. Provided by: Amazon Web Services. Located at: s3.amazonaws.com/figures.boundless.com/5101a2b3e4b04253d8aba44b/vectordecomp.jpg. License: CC BY: Attribution Provided by: Wikimedia. Located at: http://upload.wikimedia.org/Wikipedia/commons/thumb/5/5d/Position_vector.svg/220px-Position_vector.svg.png. License: CC BY: Attribution The difference between Vectors and Scalars, Introduction and Basics. Located at: http://www.youtube.com/watch? v=bap6XjDDE3k. License: Public Domain: No Known Copyright. License Terms: Standard YouTube license OpenStax College, Vector Addition and Subtraction: Graphical Methods. January 24, 2013. Provided by: OpenStax CNX. Located at: http://cnx.org/content/m42127/latest/Figure_03_02_03.jpg. License: CC BY: Attribution Vector Addition Lesson 1 of 2: Head to Tail Addition Method. Located at: http://www.youtube.com/watch?v=7puxbu24AM. License: Public Domain: No Known Copyright. License Terms: Standard YouTube license OpenStax College, Vector Addition and Subtraction: Graphical Methods. September 17, 2013. Provided by: OpenStax CNX. Located at: http://cnx.org/content/m42127/latest/. License: CC BY: Attribution Boundless. Provided by: Boundless Learning. Located at: www.boundless.com//physics/definition/component. License: CC BY-SA: Attribution-ShareAlike Scalars and Vectors. Located at: http://www.youtube.com/watch?v=EUrMI0DIh40. License: Public Domain: No Known Copyright. License Terms: Standard YouTube license Boundless. Provided by: Amazon Web Services. Located at: s3.amazonaws.com/figures.boundless.com/5101a2b3e4b04253d8aba44b/vectordecomp.jpg. License: CC BY: Attribution Provided by: Wikimedia. Located at: http://upload.wikimedia.org/Wikipedia/commons/thumb/5/5d/Position_vector.svg/220px-Position_vector.svg.png. License: CC BY: Attribution The difference between Vectors and Scalars, Introduction and Basics. Located at: http://www.youtube.com/watch? v=bap6XjDDE3k. License: Public Domain: No Known Copyright. License Terms: Standard YouTube license 3.2.11 https://phys.libretexts.org/@go/page/14446 OpenStax College, Vector Addition and Subtraction: Graphical Methods. January 24, 2013. Provided by: OpenStax CNX. Located at: http://cnx.org/content/m42127/latest/Figure_03_02_03.jpg. License: CC BY: Attribution Vector Addition Lesson 1 of 2: Head to Tail Addition Method. Located at: http://www.youtube.com/watch?v=7puxbu24AM. License: Public Domain: No Known Copyright. License Terms: Standard YouTube license OpenStax College, Vector Addition and Subtraction: Graphical Methods. January 24, 2013. Provided by: OpenStax CNX. Located at: http://cnx.org/content/m42127/latest/Figure_03_02_06a.jpg. License: CC BY: Attribution Vector Addition Lesson 2 of 2: How to Add Vectors by Components. Located at: http://www.youtube.com/watch? v=tvrynGECJ7k. License: Public Domain: No Known Copyright. License Terms: Standard YouTube license Sunil Kumar Singh, Scalar (Dot) Product. September 17, 2013. Provided by: OpenStax CNX. Located at: http://cnx.org/content/m14513/latest/. License: CC BY: Attribution scalar. Provided by: Wiktionary. Located at: en.wiktionary.org/wiki/scalar. License: CC BY-SA: Attribution-ShareAlike vector. Provided by: Wiktionary. Located at: en.wiktionary.org/wiki/vector. License: CC BY-SA: Attribution-ShareAlike magnitude. Provided by: Wiktionary. Located at: en.wiktionary.org/wiki/magnitude. License: CC BY-SA: AttributionShareAlike Scalars and Vectors. Located at: http://www.youtube.com/watch?v=EUrMI0DIh40. License: Public Domain: No Known Copyright. License Terms: Standard YouTube license Boundless. Provided by: Amazon Web Services. Located at: s3.amazonaws.com/figures.boundless.com/5101a2b3e4b04253d8aba44b/vectordecomp.jpg. License: CC BY: Attribution Provided by: Wikimedia. Located at: http://upload.wikimedia.org/Wikipedia/commons/thumb/5/5d/Position_vector.svg/220px-Position_vector.svg.png. License: CC BY: Attribution The difference between Vectors and Scalars, Introduction and Basics. Located at: http://www.youtube.com/watch? v=bap6XjDDE3k. License: Public Domain: No Known Copyright. License Terms: Standard YouTube license OpenStax College, Vector Addition and Subtraction: Graphical Methods. January 24, 2013. Provided by: OpenStax CNX. Located at: http://cnx.org/content/m42127/latest/Figure_03_02_03.jpg. License: CC BY: Attribution Vector Addition Lesson 1 of 2: Head to Tail Addition Method. Located at: http://www.youtube.com/watch?v=7puxbu24AM. License: Public Domain: No Known Copyright. License Terms: Standard YouTube license OpenStax College, Vector Addition and Subtraction: Graphical Methods. January 24, 2013. Provided by: OpenStax CNX. Located at: http://cnx.org/content/m42127/latest/Figure_03_02_06a.jpg. License: CC BY: Attribution Vector Addition Lesson 2 of 2: How to Add Vectors by Components. Located at: http://www.youtube.com/watch? v=tvrynGECJ7k. License: Public Domain: No Known Copyright. License Terms: Standard YouTube license Sunil Kumar Singh, Scalar (Dot) Product. March 12, 2014. Provided by: OpenStax CNX. Located at: http://cnx.org/content/m14513/latest/vm2a.gif. License: CC BY: Attribution Unit vector. Provided by: Wikipedia. Located at: en.Wikipedia.org/wiki/Unit_vector. License: CC BY-SA: AttributionShareAlike Scalar (physics). Provided by: Wikipedia. Located at: en.Wikipedia.org/wiki/Scalar_(physics). License: CC BY-SA: Attribution-ShareAlike Sunil Kumar Singh, Scalar (Dot) Product. September 17, 2013. Provided by: OpenStax CNX. Located at: http://cnx.org/content/m14513/latest/. License: CC BY: Attribution scalar. Provided by: Wiktionary. Located at: en.wiktionary.org/wiki/scalar. License: CC BY-SA: Attribution-ShareAlike unit vector. Provided by: Wiktionary. Located at: en.wiktionary.org/wiki/unit_vector. License: CC BY-SA: AttributionShareAlike Scalars and Vectors. Located at: http://www.youtube.com/watch?v=EUrMI0DIh40. License: Public Domain: No Known Copyright. License Terms: Standard YouTube license Boundless. Provided by: Amazon Web Services. Located at: s3.amazonaws.com/figures.boundless.com/5101a2b3e4b04253d8aba44b/vectordecomp.jpg. License: CC BY: Attribution Provided by: Wikimedia. Located at: http://upload.wikimedia.org/Wikipedia/commons/thumb/5/5d/Position_vector.svg/220px-Position_vector.svg.png. License: CC BY: Attribution The difference between Vectors and Scalars, Introduction and Basics. Located at: http://www.youtube.com/watch? v=bap6XjDDE3k. License: Public Domain: No Known Copyright. License Terms: Standard YouTube license 3.2.12 https://phys.libretexts.org/@go/page/14446 OpenStax College, Vector Addition and Subtraction: Graphical Methods. January 24, 2013. Provided by: OpenStax CNX. Located at: http://cnx.org/content/m42127/latest/Figure_03_02_03.jpg. License: CC BY: Attribution Vector Addition Lesson 1 of 2: Head to Tail Addition Method. Located at: http://www.youtube.com/watch?v=7puxbu24AM. License: Public Domain: No Known Copyright. License Terms: Standard YouTube license OpenStax College, Vector Addition and Subtraction: Graphical Methods. January 24, 2013. Provided by: OpenStax CNX. Located at: http://cnx.org/content/m42127/latest/Figure_03_02_06a.jpg. License: CC BY: Attribution Vector Addition Lesson 2 of 2: How to Add Vectors by Components. Located at: http://www.youtube.com/watch? v=tvrynGECJ7k. License: Public Domain: No Known Copyright. License Terms: Standard YouTube license Sunil Kumar Singh, Scalar (Dot) Product. March 12, 2014. Provided by: OpenStax CNX. Located at: http://cnx.org/content/m14513/latest/vm2a.gif. License: CC BY: Attribution Sunil Kumar Singh, Scalar (Dot) Product. January 16, 2015. Provided by: OpenStax CNX. Located at: http://cnx.org/content/m14513/latest/vm2a.gif. License: CC BY: Attribution Displacement (vector). Provided by: Wikipedia. Located at: en.Wikipedia.org/wiki/Displacement_(vector). License: CC BY-SA: Attribution-ShareAlike displacement. Provided by: Wiktionary. Located at: en.wiktionary.org/wiki/displacement. License: CC BY-SA: AttributionShareAlike acceleration. Provided by: Wiktionary. Located at: en.wiktionary.org/wiki/acceleration. License: CC BY-SA: AttributionShareAlike velocity. Provided by: Wiktionary. Located at: en.wiktionary.org/wiki/velocity. License: CC BY-SA: Attribution-ShareAlike Scalars and Vectors. Located at: http://www.youtube.com/watch?v=EUrMI0DIh40. License: Public Domain: No Known Copyright. License Terms: Standard YouTube license Boundless. Provided by: Amazon Web Services. Located at: s3.amazonaws.com/figures.boundless.com/5101a2b3e4b04253d8aba44b/vectordecomp.jpg. License: CC BY: Attribution Provided by: Wikimedia. Located at: http://upload.wikimedia.org/Wikipedia/commons/thumb/5/5d/Position_vector.svg/220px-Position_vector.svg.png. License: CC BY: Attribution The difference between Vectors and Scalars, Introduction and Basics. Located at: http://www.youtube.com/watch? v=bap6XjDDE3k. License: Public Domain: No Known Copyright. License Terms: Standard YouTube license OpenStax College, Vector Addition and Subtraction: Graphical Methods. January 24, 2013. Provided by: OpenStax CNX. Located at: http://cnx.org/content/m42127/latest/Figure_03_02_03.jpg. License: CC BY: Attribution Vector Addition Lesson 1 of 2: Head to Tail Addition Method. Located at: http://www.youtube.com/watch?v=7puxbu24AM. License: Public Domain: No Known Copyright. License Terms: Standard YouTube license OpenStax College, Vector Addition and Subtraction: Graphical Methods. January 24, 2013. Provided by: OpenStax CNX. Located at: http://cnx.org/content/m42127/latest/Figure_03_02_06a.jpg. License: CC BY: Attribution Vector Addition Lesson 2 of 2: How to Add Vectors by Components. Located at: http://www.youtube.com/watch? v=tvrynGECJ7k. License: Public Domain: No Known Copyright. License Terms: Standard YouTube license Sunil Kumar Singh, Scalar (Dot) Product. March 12, 2014. Provided by: OpenStax CNX. Located at: http://cnx.org/content/m14513/latest/vm2a.gif. License: CC BY: Attribution Sunil Kumar Singh, Scalar (Dot) Product. January 16, 2015. Provided by: OpenStax CNX. Located at: http://cnx.org/content/m14513/latest/vm2a.gif. License: CC BY: Attribution Boundless. Provided by: Amazon Web Services. Located at: s3.amazonaws.com/figures.boundless.com/510a0e5de4b0f11e4bcb01ad/Man_walking_up_a_hill.png. License: CC BY: Attribution This page titled 3.2: Vectors is shared under a not declared license and was authored, remixed, and/or curated by Boundless. 3.2.13 https://phys.libretexts.org/@go/page/14446 3.3: Projectile Motion Basic Equations and Parabolic Path Projectile motion is a form of motion where an object moves in parabolic path; the path that the object follows is called its trajectory. learning objectives Assess the effect of angle and velocity on the trajectory of the projectile; derive maximum height using displacement Projectile Motion Projectile motion is a form of motion where an object moves in a bilaterally symmetrical, parabolic path. The path that the object follows is called its trajectory. Projectile motion only occurs when there is one force applied at the beginning on the trajectory, after which the only interference is from gravity. In a previous atom we discussed what the various components of an object in projectile motion are. In this atom we will discuss the basic equations that go along with them in the special case in which the projectile initial positions are null (i.e. x = 0 and y = 0 ). 0 0 Initial Velocity The initial velocity can be expressed as x components and y components: ux = u ⋅ cos θ (3.3.1) uy = u ⋅ sin θ (3.3.2) In this equation, u stands for initial velocity magnitude and θ refers to projectile angle. Time of Flight The time of flight of a projectile motion is the time from when the object is projected to the time it reaches the surface. As we discussed previously, T depends on the initial velocity magnitude and the angle of the projectile: 2 ⋅ uy T= (3.3.3) g 2 ⋅ u ⋅ sin θ T= (3.3.4) g Acceleration In projectile motion, there is no acceleration in the horizontal direction. The acceleration, a, in the vertical direction is just due to gravity, also known as free fall: ax = 0 (3.3.5) ay = −g (3.3.6) Velocity The horizontal velocity remains constant, but the vertical velocity varies linearly, because the acceleration is constant. At any time, t , the velocity is: ux = u ⋅ cos θ (3.3.7) uy = u ⋅ sin θ − g ⋅ t (3.3.8) You can also use the Pythagorean Theorem to find velocity: − −−−− − 2 2 u = √ ux + uy (3.3.9) Displacement At time, t, the displacement components are: 3.3.1 https://phys.libretexts.org/@go/page/14447 x = u ⋅ t ⋅ cos θ (3.3.10) 1 y = u ⋅ t ⋅ sin θ − 2 gt (3.3.11) 2 − − − − − − 2 2 +y The equation for the magnitude of the displacement is Δr = √x . Parabolic Trajectory We can use the displacement equations in the x and y direction to obtain an equation for the parabolic form of a projectile motion: g y = tan θ ⋅ x − 2 2 2 2⋅u ⋅ cos ⋅x (3.3.12) θ Maximum Height The maximum height is reached when v object to reach maximum height y =0 . Using this we can rearrange the velocity equation to find the time it will take for the u ⋅ sin θ th = (3.3.13) g where t stands for the time it takes to reach maximum height. From the displacement equation we can find the maximum height h 2 u 2 ⋅ sin θ h= (3.3.14) 2⋅g Range The range of the motion is fixed by the condition range of the motion: y =0 . Using this we can rearrange the parabolic motion equation to find the 2 u ⋅ sin 2θ R = . (3.3.15) g Range of Trajectory: The range of a trajectory is shown in this figure. 3.3.2 https://phys.libretexts.org/@go/page/14447 Projectiles at an Angle - Very Clear Explanation Projectiles at an Angle: This video gives a clear and simple explanation of how to solve a problem on Projectiles Launched at an Angle. I try to go step by step through this difficult problem to layout how to solve it in a super clear way. 2D kinematic problems take time to solve, take notes on the order of how I solved it. Best wishes. Tune into my other videos for more help. Peace. Solving Problems In projectile motion, an object moves in parabolic path; the path the object follows is called its trajectory. learning objectives Identify which components are essential in determining projectile motion of an object We have previously discussed projectile motion and its key components and basic equations. Using that information, we can solve many problems involving projectile motion. Before we do this, let’s review some of the key factors that will go into this problemsolving. What is Projectile Motion? Projectile motion is when an object moves in a bilaterally symmetrical, parabolic path. The path that the object follows is called its trajectory. Projectile motion only occurs when there is one force applied at the beginning, after which the only influence on the trajectory is that of gravity. What are the Key Components of Projectile Motion? The key components that we need to remember in order to solve projectile motion problems are: Initial launch angle, θ 3.3.3 https://phys.libretexts.org/@go/page/14447 Initial velocity, u Time of flight, T Acceleration, a Horizontal velocity, v Vertical velocity, v Displacement, d Maximum height, H Range, R x y How To Solve Any Projectile Motion Problem (The T… T… How To Solve Any Projectile Motion Problem (The Toolbox Method): Introducing the “Toolbox” method of solving projectile motion problems! Here we use kinematic equations and modify with initial conditions to generate a “toolbox” of equations with which to solve a classic three-part projectile motion problem. Now, let’s look at two examples of problems involving projectile motion. Example 3.3.1: Example 1 Let’s say you are given an object that needs to clear two posts of equal height separated by a specific distance. Refer to for this – example. The projectile is thrown at 25√2 m/s at an angle of 45°. If the object is to clear both posts, each with a height of 30m, find the minimum: (a) position of the launch on the ground in relation to the posts and (b) the separation between the posts. For simplicity’s sake, use a gravity constant of 10. Problems of any type in physics are much easier to solve if you list the things that you know (the “givens”). 3.3.4 https://phys.libretexts.org/@go/page/14447 Diagram for Example 1: Use this figure as a reference to solve example 1. The problem is to make sure the object is able to clear both posts. Solution: The first thing we need to do is figure out at what time tt the object reaches the specified height. Since the motion is in a parabolic shape, this will occur twice: once when traveling upward, and again when the object is traveling downward. For this we can use the equation of displacement in the vertical direction, y − y : 0 1 y − y0 = (vy ⋅ t) − ( 2 ⋅g⋅t ) (3.3.16) 2 We substitute in the appropriate variables: m –m ∘ vy = u ⋅ sin θ = 25 √2 ⋅ sin 45 = 25 s s (3.3.17) Therefore: 1 30m = 25 ⋅ t − 2 ⋅ 10 ⋅ t (3.3.18) 2 We can use the quadratic equation to find that the roots of this equation are 2s and 3s. This means that the projectile will reach 30m after 2s, on its way up, and after 3s, on its way down. Example 2 An object is launched from the base of an incline, which is at an angle of 30°. If the launch angle is 60° from the horizontal and the launch speed is 10 m/s, what is the total flight time? The following information is given: u = 10 ; θ = 60°; g = 10 . m s m 2 s Diagram for Example 2: When dealing with an object in projectile motion on an incline, we first need to use the given information to reorient the coordinate system in order to have the object launch and fall on the same surface. Solution: In order to account for the incline angle, we have to reorient the coordinate system so that the points of projection and return are on the same level. The angle of projection with respect to the x direction is θ − α , and the acceleration in the y direction is g ⋅ cos α . We replace θ with θ − α and g with g ⋅ cos α : 2 ⋅ u ⋅ sin(θ) T = 2 ⋅ u ⋅ sin(θ − α) = g 2 ⋅ 10 ⋅ sin(60 − 30) = g ⋅ cos(α) 20 ⋅ sin(30) = 10 ⋅ cos(30) (3.3.19) 10 ⋅ cos(30) 2 T = – √3 s (3.3.20) 3.3.5 https://phys.libretexts.org/@go/page/14447 Zero Launch Angle An object launched horizontally at a height H travels a range v − − − 0√ 2H g − − − during a time of flight T = √ 2H g . learning objectives Explain the relationship between the range and the time of flight Projectile motion is a form of motion where an object moves in a parabolic path. The path followed by the object is called its trajectory. Projectile motion occurs when a force is applied at the beginning of the trajectory for the launch (after this the projectile is subject only to the gravity). One of the key components of the projectile motion, and the trajectory it follows, is the initial launch angle. The angle at which the object is launched dictates the range, height, and time of flight the object will experience while in projectile motion. shows different paths for the same object being launched at the same initial velocity and different launch angles. As illustrated by the figure, the larger the initial launch angle and maximum height, the longer the flight time of the object. Projectile Trajectories: The launch angle determines the range and maximum height that an object will experience after being launched.This image shows that path of the same object being launched at the same speed but different angles. We have previously discussed the effects of different launch angles on range, height, and time of flight. However, what happens if there is no angle, and the object is just launched horizontally? It makes sense that the object should be launched at a certain height ( H ), otherwise it wouldn’t travel very far before hitting the ground. Let’s examine how an object launched horizontally at a height H travels. In our case is when α is 0. Y α X g Projectile motion: Projectile moving following a parabola.Initial launch angle is αα, and the velocity is v . 0 Duration of Flight There is no vertical component in the initial velocity (v ) because the object is launched horizontally. Since the object travels distance H in the vertical direction before it hits the ground, we can use the kinematic equation for the vertical motion: 0 3.3.6 https://phys.libretexts.org/@go/page/14447 1 (y − y0 ) = −H = 0 ⋅ T − 2 gT (3.3.21) 2 Here, T is the duration of the flight before the object its the ground. Therefore: − − − 2H T =√ (3.3.22) g Range In the horizontal direction, the object travels at a constant speed direction) is given as: v0 during the flight. Therefore, the range R (in the horizontal − − − 2H R = v0 ⋅ T = v0 √ (3.3.23) g General Launch Angle The initial launch angle (0-90 degrees) of an object in projectile motion dictates the range, height, and time of flight of that object. learning objectives Choose the appropriate equation to find range, maximum height, and time of flight Projectile motion is a form of motion where an object moves in a bilaterally symmetrical, parabolic path. The path that the object follows is called its trajectory. Projectile motion only occurs when there is one force applied at the beginning of the trajectory, after which the only interference is from gravity. One of the key components of projectile motion and the trajectory that it follows is the initial launch angle. This angle can be anywhere from 0 to 90 degrees. The angle at which the object is launched dictates the range, height, and time of flight it will experience while in projectile motion. shows different paths for the same object launched at the same initial velocity at different launch angles. As you can see from the figure, the larger the initial launch angle, the closer the object comes to maximum height and the longer the flight time. The largest range will be experienced at a launch angle up to 45 degrees. Launch Angle: The launch angle determines the range and maximum height that an object will experience after being launched. This image shows that path of the same object being launched at the same velocity but different angles. The range, maximum height, and time of flight can be found if you know the initial launch angle and velocity, using the following equations: 2 v R = i sin 2 θi (3.3.24) g 2 v h = i 2 sin θi (3.3.25) 2g T = 2 vi sin θ (3.3.26) g Where R – Range, h – maximum height, T – time of flight, vi – initial velocity, θi – initial launch angle, g – gravity. 3.3.7 https://phys.libretexts.org/@go/page/14447 Now that we understand how the launch angle plays a major role in many other components of the trajectory of an object in projectile motion, we can apply that knowledge to making an object land where we want it. If there is a certain distance, d, that you want your object to go and you know the initial velocity at which it will be launched, the initial launch angle required to get it that distance is called the angle of reach. It can be found using the following equation: 1 θ = −1 sin 2 gd ( 2 ) (3.3.27) v Key Points: Range, Symmetry, Maximum Height Projectile motion is a form of motion where an object moves in parabolic path. The path that the object follows is called its trajectory. learning objectives Construct a model of projectile motion by including time of flight, maximum height, and range What is Projectile Motion ? Projectile motion is a form of motion where an object moves in a bilaterally symmetrical, parabolic path. The path that the object follows is called its trajectory. Projectile motion only occurs when there is one force applied at the beginning on the trajectory, after which the only interference is from gravity. In this atom we are going to discuss what the various components of an object in projectile motion are, we will discuss the basic equations that go along with them in another atom, “Basic Equations and Parabolic Path” Key Components of Projectile Motion: Time of Flight, T: The time of flight of a projectile motion is exactly what it sounds like. It is the time from when the object is projected to the time it reaches the surface. The time of flight depends on the initial velocity of the object and the angle of the projection, θθ. When the point of projection and point of return are on the same horizontal plane, the net vertical displacement of the object is zero. Symmetry: All projectile motion happens in a bilaterally symmetrical path, as long as the point of projection and return occur along the same horizontal surface. Bilateral symmetry means that the motion is symmetrical in the vertical plane. If you were to draw a straight vertical line from the maximum height of the trajectory, it would mirror itself along this line. Maximum Height, H: The maximum height of a object in a projectile trajectory occurs when the vertical component of velocity, vyvy, equals zero. As the projectile moves upwards it goes against gravity, and therefore the velocity begins to decelerate. Eventually the vertical velocity will reach zero, and the projectile is accelerated downward under gravity immediately. Once the projectile reaches its maximum height, it begins to accelerate downward. This is also the point where you would draw a vertical line of symmetry. Range of the Projectile, R: The range of the projectile is the displacement in the horizontal direction. There is no acceleration in this direction since gravity only acts vertically. shows the line of range. Like time of flight and maximum height, the range of the projectile is a function of initial speed. 3.3.8 https://phys.libretexts.org/@go/page/14447 Range: The range of a projectile motion, as seen in this image, is independent of the forces of gravity. Key Points Objects that are projected from, and land on the same horizontal surface will have a vertically symmetrical path. The time it takes from an object to be projected and land is called the time of flight. This depends on the initial velocity of the projectile and the angle of projection. When the projectile reaches a vertical velocity of zero, this is the maximum height of the projectile and then gravity will take over and accelerate the object downward. The horizontal displacement of the projectile is called the range of the projectile, and depends on the initial velocity of the object. When solving problems involving projectile motion, we must remember all the key components of the motion and the basic equations that go along with them. Using that information, we can solve many different types of problems as long as we can analyze the information we are given and use the basic equations to figure it out. To clear two posts of equal height, and to figure out what the distance between these posts is, we need to remember that the trajectory is a parabolic shape and that there are two different times at which the object will reach the height of the posts. When dealing with an object in projectile motion on an incline, we first need to use the given information to reorientate the coordinate system in order to have the object launch and fall on the same surface. For the zero launch angle, there is no vertical component in the initial velocity. − − − The duration of the flight before the object hits the ground is given as T = √ 2H g . In the horizontal direction, the object travels at a constant speed v0 during the flight. The range R (in the horizontal direction) is − − − given as: R = v 0 2H ⋅ T =