Chapter 03 Motion in One Dimension - Physics PDF

## Chapter 03 Motion in One Dimension ### Inside 1. Rest and motion * Some basic terms related to motion 2. Kinematic equations for uniformly accelerated motion 3. Motion under gravity * Equations for motion under gravity 4. Non-uniformly accelerated motion 5. Graphical representa...

## Chapter 03 Motion in One Dimension ### Inside 1. Rest and motion * Some basic terms related to motion 2. Kinematic equations for uniformly accelerated motion 3. Motion under gravity * Equations for motion under gravity 4. Non-uniformly accelerated motion 5. Graphical representation of motion 6. Relative velocity * Different cases of relative velocity * Examples of relative motion 7. Examples of cases of relative motion ### Motion is defined as the change in position of an object with time. - When the object moves along a single axis, the motion is known as one dimensional motion and such a motion is along a straight line only, which may be horizontal, vertical or slanted. ### Frame of reference - A system of coordinate axes which defines the position of a particle or an event in two or three-dimensional space is called a frame of reference. - A simple word, a frame of reference is the frame in which an observer sits and makes observations. ### Types of frame of reference Frame of references are of two types. 1. **Inertial frame of reference:** A non-accelerating frame of reference is called an inertial frame of reference. A frame of reference which is either at rest or moving with a constant velocity is an inertial frame of reference. 2. **Non-inertial frame of reference:** An accelerating frame of reference is called a non-inertial frame of reference. If an observer is sit in an accelerated moving frame, then the frame is called a non-inertial frame of reference. ### Rest and motion - If the position of an object does not change with respect to its surrounding with the passage of time, it is said to be at rest. (e.g., Book lying on the table, a person sitting on a chair) - If the position of an object is continuously changing with respect to its surrounding, then it is said to be in the state of motion. (e.g., The walking man, crawling insects, water flowing down a dam) ### Rest and motion are relative terms - Rest and motion are always relative but never absolute. It means an object can be at rest for an observer, but the same object can be in motion when observed by other observer (e.g., a person sitting in his house is at rest with respect to earth but in motion with respect to the moon.) ### Classification of motion On the basis of the number of coordinates required to specify the position of the object, motion of an object can be classified as: 1. **One-dimensional motion:** The motion of an object is considered as one dimensional if only one coordinate is needed to specify the position of the object. * In one-dimensional motion, the object moves along a straight line. (e.g., A boy running on a straight line, motion of freely falling body) 2. **Two-dimensional motion:** The motion of an object is considered as two dimensional if two coordinates are needed to specify the position of the object. * In two-dimensional motion, the object moves in a plane. (e.g., A satellite revolving around the earth, motion of a billiard ball) 3. **Three-dimensional motion:** The motion of an object is considered as three-dimensional if all the three coordinates are needed to specify the position of the object. * In three-dimensional motion, the object moves in a space. (e.g., Butterfly flying in a garden, the motion of water molecules) ### Some basic terms related to motion 1. **Point object:** An object is considered as a point object if the size of the object is much smaller than the distance it moves in a reasonable duration of time. (e.g., Earth can be considered as a point object during its revolution around sun because it covers the much larger distance than its own size.) 2. **Distance and Displacement:** * **Distance:** The length of the path covered by the object in a given time-interval is known as its distance. Distance is a scalar quantity. The unit of distance is the meter in SI or MKS and centimeter in CGS. Its dimensional formula is [M°LT° ]. * **Displacement:** The change in position vector of an object in a particular direction is termed as displacement. In other words, displacement is the shortest distance between the initial and final position of the moving object. If x₁ and x₂ are the initial and final position vectors of the moving object, then the displacement of the particle is given by: * Δx = x₂ -x₁ * If x₂>x₁, then Δx is positive. * If x₁> x₂, then Δx is negative. * If x₁ = x₂, then Δx is zero. * Displacement is a vector quantity as it possesses both, the magnitude and direction. * The unit of displacement is the meter in SI or MKS and centimeter in CGS. * Its dimensional formula is [MOLT° ]. ### 3. Speed - The time rate of distance travelled by an object in any direction is called speed of the object. * Speed (v) = Distance travelled / Time taken * It is a scalar quantity. * The unit of speed in SI or MKS system is ms -1 and in CGS system it cms¯¹. * Its dimensional formula is [M°LT¯¹]. * For a moving body, speed is always positive and can never be negative or zero. ### Average speed - The ratio of the total distance travelled by the object to the total time taken is called the average speed of the object. * Average speed = Total distance travelled / Total time taken ### Instantaneous speed - The speed of a particle at any instant of time is known as its instantaneous speed. * Instantaneous speed = lim Δt→0 Δs / Δt = ds / dt * where s represents distance. ### 4. Velocity - The time rate of change of position or displacement of an object with time is called the velocity of that object. * Velocity = Displacement / Time * Velocity is a vector quantity. * The unit of velocity in SI or MKS system is ms¯¹ and in CGS system it is cms¯¹. * Its dimensional formula is [MOLT¯¹]. * The velocity of an object is taken to be positive if the object is moving towards the right of the origin and is taken to be negative if the object is moving towards the left of the origin. ### Velocity vs Speed 1. Velocity of an object can be changed by changing the object's speed or direction of motion or both. 2. For an object in a time interval (t): |Velocity| ≤ Speed i.e., the magnitude of velocity of an object is always equal to or less than its speed. 3. If a body is moving in a straight line, then the magnitude of its speed and velocity will be equal. 4. Average velocity could be zero or positive or negative but average speed is always positive for a moving body. ### Average velocity - The ratio of the total displacement to the total time taken is called the average velocity. * Average velocity = Total displacement (Δx) / Total time (Δt) * If velocity of the object changes at a uniform rate, then * Average velocity = (Initial velocity + Final velocity) / 2 ### Instantaneous velocity - The velocity of a particle at any instant of time is known as its instantaneous velocity. * v = lim Δt→0 Δx / Δt = dx / dt ### 5. Acceleration - The time rate of change of velocity of a body is called acceleration. * Acceleration = change in velocity (Δv)/ Time interval (Δt) * If in a given time interval the velocity of a body changes from u to v, then acceleration a is expressed as * a = (Final velocity - Initial velocity)/ Time interval= v - u / t * It is a vector quantity, denoted by a and its SI unit is ms ̄² and CGS is cms¯². * Its dimensional formula is [MLT¯²]. * Its direction is same as that of change in velocity (not of the velocity). ### Key points regarding acceleration motion 1. A body falling down from a height or a body rolling down on a smooth inclined plane, has uniform acceleration. 2. If a car travelling along a straight road increases its speed by unequal amounts in equal intervals of time, then the car is said to be moving with non-uniform acceleration. 3. The acceleration is created by the accelerator of the vehicles and the applications of breaks give uniform deceleration to the vehicles. The acceleration produced in spring-block system is non-uniform acceleration. 4. Instantaneous acceleration of an object * is equal to the first time derivative of velocity at the given instant. * is equal to the second time derivative of position of the object at the given instant. 5. If a particle is accelerated for time t₁ with acceleration a₁, and for time t₂ with acceleration a₂, then the average acceleration is: * aav = (a₁t₁ + a₂t₂) / (t₁ + t₂) 6. Acceleration can also be written as: * a = dv / dt = dv / dx x dx / dt = v x dv / dx ### Instantaneous acceleration - The acceleration of an object at a given instant of time or at a given point during the motion is called its instantaneous acceleration. * a = lim Δt→0 Δv / Δt = dv / dt ### Retardation - When the velocity of a body increases with time, acceleration is positive and when the velocity of a body decreases with time (i.e., u > v), then acceleration becomes negative. Negative acceleration is also called deceleration or retardation. ### Average acceleration - When an object is moving with a variable acceleration, then the average acceleration of the object for the given motion is defined as the ratio of the total change in velocity of the object during motion to the total time taken. * Average acceleration = Total change in velocity / Total time taken ### Kinematic equations for uniformly accelerated motion - When a body is moving along a straight line with uniform acceleration, its motion is called uniformly accelerated motion. We can establish the relation between velocity of the body, acceleration of the body, and the distance travelled by the body in a particular time interval by a set of equations. These equations are known as kinematic equations or equations of motion. * The three equations of motion on a straight line are: * v = u + at * s = ut + 1/2 at² * v² - u² = 2as * where u is the initial velocity of the body, v is the final velocity of the body after t second, a is the uniform acceleration of the body, and s is the distance travelled in this time. * Distance travelled in nth second: sn = u + 1/2 a(2n -1) ### Motion under gravity - The objects falling towards the earth under the gravitational force alone, are called freely falling objects and such fall is called free fall. - Whenever an object falls towards the earth, an acceleration is involved, this acceleration is due to the earth’s gravitational pull and is called acceleration due to gravity. - The value of acceleration due to gravity near the earth surface is 9.8 ms¯². - It is independent of the mass of freely falling objects and is denoted by g. ### Equations for motion under gravity - When the objects fall under the influence of gravity, its motion is uniformly accelerated motion. Hence, equations of motion are applicable in this case also. - The equations for motion under gravity are given below: 1. If the particle is projected vertically upwards: * v = u - gt * h = ut - 1/2 gt² * v² = u² - 2gh 2. If the particle is projected vertically downward with some velocity from some height: * v = u + gt * h = ut + 1/2 gt² * v² = u² + 2gh ### Instantaneous acceleration - The acceleration of an object at a given instant of time or at a given point during the motion, is called its instantaneous acceleration. * a = lim Δt→0 Δv / Δt = dv / dt ### Non-uniformly accelerated motion - When acceleration of particle is not constant, motion is known as non-uniformly accelerated motion. Then basic equations of velocity and acceleration can be written as * v = ds / dt or sometimes v = dr / dt (ii) a = dv / dt * (iii) ds = v dt (iv) dv = adt * For one dimensional motion, above relations can be written as: * (i) v = ds / dt (ii) a = dv / dt = v x dv/ ds (iii) ds = v dt and (iv) dv = adt or vdv = ads * Such problems can be solved either by differentiation or integration (with some boundary conditions). ### Graphical representation of motion - Motion of a point or body or a particle in all aspects can be shown with the help of the graph, such as displacement-time graph and velocity-time graph etc. - The tabular forms of displacement-time and velocity-time graphs are given for one-dimensional motion. 1. **Displacement-time graph:** * Displacement-time graph gives the instantaneous value of displacement at any instant. * The slope of the tangent drawn to the graph at any instant of time gives the instantaneous velocity at that instant. * The s-t graph cannot make sharp turns. 2. **Velocity-time graph:** * Velocity-time graph gives the instantaneous value of velocity at any instant. * The slope of the tangent drawn on graph gives instantaneous acceleration. * Area under v-t graph with the time axis gives the value of displacement covered in the given time. * The v-t curve cannot take sharp turns. 3. **Acceleration-time graph:** * The area of the a-t graph between time t₁ to t₂ gives the change in velocity. ### Relative velocity - The term relative is frequently used for comparison of displacement, velocity and acceleration of the two objects. - The time rate of change of relative position of one object with respect to another is called relative velocity. * Let two objects A and B are moving along the + ve direction on X-axis. At time t, their displacement from the origin be XA and XB. * Their velocities are VA = dXA / dt and VB = dXB / dt * The displacement of B relative to A: XBA = XB - XA * Rate of change of relative displacement of B w.r.t. time t is: d(XBA) / dt = d(XB - XA) / dt = dXB / dt - dXA / dt = VB - VA * VBA = VB - VA * Similarly, relative acceleration of A with respect to B is: a AB = aA - aB * If it is a one-dimensional motion, we can treat the vectors as scalars just by assigning the positive sign to one direction and negative to the other. * So, in case of a one-dimensional motion, the above equations can be written as: * VAB = VA - VB * аАВ = аА - аВ ### Different cases of relative velocity 1. **When the two objects move with equal velocities, i.e. VA = VB Or VB - VA = 0.** It means, the two objects stay at a constant distance apart during the whole journey. In this case, the position-time graph of two objects are parallel straight lines. 2. **If both objects A and B move along parallel straight lines in the opposite direction, then the relative velocity of B w.r.t. A is given as:** VBA = V B - (-VA) = VB + VA and the relative velocity of A w.r.t. B is given by: V AB = VA - (VB) = V A + VB 3. **IF VA<VB, VA-VB is negative.** Then, x-xo < 0 ⇒ x < xo where, x = initial displacement of object A w.r.t. B and x = displacement of object A w.r.t. B at time t 4. **IF VA > VB, VA - VB is positive.** It means the separation between the two objects will go on increasing with time i.e., the separation (x-xo) between them will increase by an amount (VA-VB) after each unit of time. Therefore, their position-time graphs will open out gradually as shown below. ### Crossing the river - To cross the river over shortest distance, i.e., to cross the river straight, the man should swim upstream making an angle θ with OB such that, OB gives the direction of resultant velocity (VMR) of velocity of swimmer and velocity of river water as shown in figure. ### Drift - It is defined as the displacement of man in the direction of river flow as shown ### Examples - Example 3.64: To a man walking at the rate of 3 km/h, the rain appears to fall vertically. When he increases his speed to 6 km/h, it appears to meet him at an angle of 45° with vertical. Find the speed of the rain. - Example 3.65: A man crosses a river in a boat. If he crosses the river in minimum time, he takes 10 min with a drift of 120 m. If he crosses the river taking the shortest path, he takes 12.5 min. Find: (a) width of the river. (b) velocity of the boat with respect to water. (c) speed of the current. ### Checkpoints - Check point 3.1 - Check point 3.2 - Check point 3.3 - Check point 3.4 - Check point 3.5 - Check point 3.6 - Check point 3.7 ### Answers This document includes solutions for checkpoints and some examples from the chapter.