is moving in a positive direction with a positive acceleration. (b) An object is moving in positive direction with a negative acceleration. (c) An object is moving in negative direction with a negative acceleration. (d) An object is moving in positive direction till time t , and then turns back 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 . A general proof of this statement requires use of calculus. We can, however, see that it is true for the simple case of an object moving with constant velocity u . Its velocity-time graph is as shown in Fig.
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 = and 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. How come in this case an area is equal to a distance?
Think! Note the dimensions of quantities on the two coordinate axes, and you will arrive at the answer. 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 .
What this means physically is that acceleration and velocity cannot change values abruptly at an instant. Changes are always continuous. . KINEMATIC E Q UATIONS FOR UNIFORMLY ACCELERATED MOTION For uniformly accelerated motion, we can derive some simple equations that relate displacement ( x ), time taken ( t ), initial velocity ( v ), final velocity ( v ) and acceleration ( a ).
Equation ( . ) already obtained gives a relation between final