Chapter 3: Two Dimensional Kinematics PDF

Chapter 3 : Two dimensional Kinematics Mrs. Pooja Brahmaiahchari  Introduction Motion along a curved path on a flat surface or a plane (such as that of a ball on a pool table or a skater on an ice rink) is two-dimensional, and thus described by two- dimensional kinematics. A...

Chapter 3 : Two dimensional Kinematics Mrs. Pooja Brahmaiahchari  Introduction Motion along a curved path on a flat surface or a plane (such as that of a ball on a pool table or a skater on an ice rink) is two-dimensional, and thus described by two- dimensional kinematics. A vector quantity is a quantity that is fully described by both magnitude and direction. On the other hand, a scalar quantity is a quantity that is fully described by its magnitude. Vector addition and subtraction graphical method Vector Addition: Head-to-Tail Method A person walks 9 blocks east and then 5 blocks north. The head-to-tail method is a graphical way to add vectors. The tail of the vector is the starting point of the vector, and the head (or tip) of a vector is the final, pointed end of the arrow. Vector addition is commutative. Vectors can be added in any order. 𝐴+𝐵 =𝐵+𝐴  To get the magnitude of the resultant, measure its length with a ruler. (Note that in most calculations, we will use the Pythagorean theorem to determine this length.) To get the direction of the resultant, measure the angle it makes with the reference frame using a protractor. (Note that in most calculations, we will use trigonometric relationships to determine this angle.) Examples 1.Vector A has magnitude of 30 km and points to west Vector B has magnitude of 40 km and points south what is 𝐴Ԧ + 𝐵 2. Vector A has magnitude of 10 km and points to north Vector B has magnitude of 5 km and points west what is 𝐴Ԧ + 𝐵 Vector Subtraction To define subtraction (say we want to subtract B from A ,written A - B, we must first define what we mean by subtraction. The negative of a vector B is defined to be -B ; that is, Graphically the negative of any vector has the same magnitude but the opposite direction, Note that vector subtraction is the addition of a negative vector. The order of subtraction does not affect the results. 𝐴 − 𝐵 = 𝐴 + (−𝐵) 3. Vector A has magnitude of 6 and points to North Vector B has magnitude of 8 and points East What is the magnitude of B-A ? Multiplication of vectors and scalars To multiply a vector by a scalar, take the product of the magnitudes. When vector A is multiplied by a scalar c, the magnitude of the vector becomes the absolute value of cA, if c is positive, the direction of the vector does not change, if c is negative, the direction is reversed. Vector components A two-dimensional vector can be represented in polar coordinates (r,θ) or as in Cartesian coordinates (x,y) In order to add two or more vectors together, the vectors must be in (x,y) form. Vector Addition and Subtraction Analytical method Analytical methods of vector addition and subtraction employ geometry and simple trigonometry. Analytical methods are more concise, accurate, and precise than graphical methods The Cartesian components of a vector may be found using trigonometry. 𝐴𝑥 = 𝐴 𝑐𝑜𝑠𝜃 𝐴𝑦 = 𝐴 𝑠𝑖𝑛𝜃  We can then add the components following simple arithmetic. The final vector is found using the Pythagorean theorem and arctan. 𝐴= 𝐴2𝑥 + 𝐴2𝑦 −1 𝐴𝑦 𝜃 = tan 𝐴𝑥 PAY ATTENTION TO QUADRANT!! 4. Vector A represents the first leg of a walk in which a person walks 53.0m in a direction 20o north of east. Vector B represents the second leg, a displacement of 34.0m in a 63odirection north of east. Determine the magnitude and direction of resultant vector R. Projectile motion Projectile motion is the motion of an object thrown or projected into the air, subject to only the acceleration 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. The most important fact to remember here is that motions along perpendicular axes are independent and thus can be analyzed separately.  Since the only force acting on it is gravitational force, hence ax =0 ay=-g Both accelerations are constant, so the kinematic equations can be used. Given these assumptions, the following steps are then used to analyze projectile motion: Step 1. Resolve or break the motion into horizontal and vertical components along the x- and y-axes. Step 2. Treat the motion as two independent one-dimensional motions, one horizontal and the other vertical. Step 3. Solve for the unknowns in the two separate motions—one horizontal and one vertical. Note that the only common variable between the motions is time. Step 4: Recombine the two motions to find the total displacement and velocity. Horizontally Launched Projectiles The horizontal displacement of a projectile is dependent upon the horizontal component of the initial velocity. horizontal displacement of a projectile can be determined using the equation x = vix t 5.If a projectile has a time of flight of 8 seconds and a horizontal velocity of 20 m/s, then find the horizontal displacement. Given t = 8 s vix =20m/s x = vix t = 20 m/s 8 s = 160 m 6. A projectile is launched at a height of 30m high hill with initial horizontal velocity of 5m/s. Calculate the time it takes to reach ground and the how far it travelled horizontally. 7. A soccer ball is kicked horizontally off a 22.0-meter high hill and lands a distance of 35.0 meters from the edge of the hill. Determine the initial horizontal velocity of the soccer ball. 8. A ball is launched horizontally off a 5m high hill at initial velocity of 10m/s from the edge of the hill. a) Determine the time. b) Determine the horizontal distance of the ball. Determining the Components of a Velocity Vector The horizontal velocity component (vx) describes the influence of the velocity in displacing the projectile horizontally. The vertical velocity component (vy) describes the influence of the velocity in displacing the projectile vertically. Thus, the analysis of projectile motion problems begins by using the trigonometric methods 𝒗𝒙 = 𝒗 cos 𝜃 for x axis (horizontal) 𝒗𝒚 = 𝒗 sin 𝜃 𝒇𝒐𝒓 𝒚 𝒂𝒙𝒊𝒔 (𝒗𝒆𝒓𝒕𝒊𝒄𝒂𝒍) 9. A football leaves the ground at an angle of 30o horizontally with the initial velocity of 10 m/s. a) Determine the time b) Determine the horizontal distance. THANK YOU