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Kinematics | Representation / Equations of Motion

Reference: AP Physics Algebra, Kinematics, Equations of Motion, Equation of motion on an inclined plane, Displacement – Time, Velocity- Time & Acceleration - Time Graphs for Motion in one Dimension

 

After studying this chapter, you should be able to,

• derive equations of motion with constant acceleration;
• describe motion under gravity;
• solve numerical based on equations of motion; and understand the concept of differentiation and integration.
• know displacement – time, velocity-time & acceleration - time Graphs for Motion in one Dimension

 

EQUATION OF MOTION

Following are the three equations of motion for an object with constant acceleration. 

(a)    v = u + at 

(b)  S = ut + 1/2 at² 

(c)  v² = u² + 2 as  

Where u is the initial velocity of the body (if the body start from rest u = 0), v is the final velocity, s = displacement travelled by the body in time t seconds and a = acceleration of the body (take + sign for acceleration and – for retardation).

The displacement by the body in n^th second is given by

 

Equation of motion on an inclined plane (Kinematic)

 

Let a body of mass m slip down a plane, which is inclined at an angle q with the horizontal. If at t = 0, the body is at the top of the inclined plane, then in this case u = 0 and a = g sinq 

(i)     In this case the equations of motion are 

(a)   

                                                                       

(b)   

(c)   

 

           

(ii)    If the time taken by the body to reach the bottom is t, then 

        

        

But   

 or

        

(iii)    The velocity of the body at the bottom

                 

(iv)  The velocity of a body moving on an inclined plane does not depend on the inclination of a plane but the time taken to reach the bottom of the plane depends on the inclination of the plane. The velocity and the time taken by the body on an inclined plane depend on the height.

(v)    When a body moves on an inclined plane. It traverses one-fourth of the length of the inclined. Plane in time interval 0 to t/2 and the remaining three fourth in the time interval t/2 to t.

 

Displacement – Time, Velocity- Time & Acceleration - Time Graphs for Motion in one Dimension

 

(i)Variation of displacement (x), velocity (v) and acceleration (a) with respect to time for different types of motion. 

 

 

 

(ii)    Displacement calculation from Velocity - Time Graphs

The displacement during an interval between time t_i and t_f is the area bounded by the velocity curve and the two vertical lines t = t_i and t = t_f, as shown in figures (a) and (b).

 

(iii) Velocity calculation from Acceleration - time Graphs

Given an acceleration–versus–time graph, the change in velocity between t = t_i and t = t_f is the area bounded by the acceleration curve and the vertical lines t = t_i and t = t_f

 

 

Example 1.

A ball is projected with a velocity of 20 m/s vertically. Find the distance travelled in the first three seconds. (Use g = 10m/sec²⁾

Solution:                                                          

The problem here is to find the distance. We can calculate that the direction of the ball is changed at t = 2s. (From v = u + at, since v = 0 at highest point therefore 0=20 - 10t Þ t = 2s)

Distance travelled in first two seconds (

 Distance = Displacement,          because velocity does not change direction in one dimension)

        

         = 40 – 20 = 20 m (upward)

               Distance travelled in the next second

        

         So total distance travelled by the ball in the first three seconds

         = 20 +5 = 25m

 

Key Points:

• Following are the three equations of motion for an object with constant acceleration. 

          (a)

 

          (b)

          (c)

 

• The velocity of a body moving on an inclined plane does not depend on the inclination of a plane but the time taken to reach the bottom of the plane depends on the inclination of the plane. The velocity and the time taken by the body on an inclined plane depend on the height.

• When a body moves on an inclined plane. It traverses one-fourth of the length of the inclined. Plane in time interval 0 to t/2 and the remaining three fourth in the time interval t/2 to t.