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Acceleration

Acceleration is defined as the rate of change of velocity. When the velocity of an object increases, it is called positive acceleration. When the velocity of an object decreases, it is termed as negative acceleration.

The studies of motion of freely falling objects and motion of objects on an inclined plane, Galileo concluded that the rate of change of velocity with time is a constant of motion for all objects in free fall.

On the other hand, the change in velocity with distance is not constant - it decreases with increasing distance of fall. This led to the concept of acceleration as the rate of change of velocity with time.

The average acceleration over a time interval is defined as the change of velocity divided by the time interval:
 
where are the instantaneous velocities or simply velocities at time t2 and t1. It is the average change of velocity per unit time. The SI unit of acceleration is ms-2.

On a plot of velocity versus time, the average acceleration is the slope of straight line connecting the points corresponding to . The average acceleration for velocity-time graph shown in the figure for different time intervals 0s-10s, 10s-18s and 18s-20s are:

Instantaneous acceleration is defined in the same way as the instantaneous velocity:


The acceleration at an instant is the slope of the tangent to the v-t curve at that instant. For the curve shown in figure. We can obtain acceleration at every instant of time. The resulting a-t curve is shown in figure. We see that the acceleration is non-uniform over the period 0s to 10s. It is zero between 10s and 18s and is constant with value -12 m s-2 between 18s and 20s. When the acceleration is uniform, obviously it equals the average acceleration over that period.

Since velocity is a quantity having both magnitude and direction, a change in velocity may involve either or both of these factors. Acceleration, therefore, may result from a change in speed (magnitude), a change in direction or a change in both.

Like velocity, acceleration can also be positive, negative or zero. Position-time graphs for motion with positive, negative and zero acceleration are shown in the given figures (a), (b) and (c), respectively.

Note that the graph curves upward for positive acceleration, downward for negative acceleration and it is straight line for zero acceleration.

If the velocity of an object is at ,
 
Hence we can write,
Let us see how velocity-time graph looks like for some simple cases. The given figure shows velocity-time graph for motion with constant acceleration for the following cases:
  1. An object is moving in positive direction with a positive acceleration, for example the motion of the car in the figure between t=0s and t=10s.
  2. An object is moving in positive direction with a negative acceleration, for example, motion of the car in the figure between t=18s to 20s.
  3. An object is moving in negative direction with a negative acceleration, for example the motion of a car moving from O in the figure in negative x-direction with increasing speed.
  4. An object is moving in positive direction till time t1, and then turns back with the same negative acceleration, for example the motion of a car from point O to point Q in Fig till time t1 with decreasing speed and turning back and moving with the same negative acceleration.
An interesting feature of a velocity-time graph for any moving object is that the area under the curve represents the displacement over a given time interval. Its velocity-time graph is as shown in the figure.

 Area under v-t curve equals displacement of the object over a given time interval.

The v-t curve is a straight line parallel to the time axis and the area under it between t=0 to t=T is the area of the rectangle of height u and base T. Therefore, area = u × T = uT which is the displacement in this time interval.

Note that the x-t, v-t and a-t graphs shown in several figures in this chapter have sharp kinks at some points implying that the functions are not differentiable at these points. In any realistic situation, the functions will be differentiable at all points and the graphs will be smooth.




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