On a plot of velocity versus time, the average acceleration is the slope of the straight line connecting the points corresponding to ( v , t ) and ( v , t ). Fig. . Velocity–time graph for motions with constant acceleration.
(a) Motion in positive direction with positive acceleration, (b) Motion in positive direction with negative acceleration, (c) Motion in negative direction with negative acceleration, (d) Motion of an object with negative acceleration that changes direction at time t . Between times to t , it moves in positive x - direction and between t and t it moves in the opposite direction . Instantaneous acceleration is defined in the same way as the instantaneous velocity : lim ( . ) The acceleration at an instant is the slope of the tangent to the v–t curve at that instant.
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 changes 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 Figs.
. (a), (b) and (c), respectively. Note that the graph curves upward for positive acceleration; downward for negative acceleration and it is a straight line for zero acceleration. Although acceleration can vary with time, our study in this chapter will be restricted to motion with constant acceleration.
In this case, the average acceleration equals the constant value of acceleration during the interval. If the velocity of an object is v o at t = and v at time t , we have or , v a t ( . ) Fig. .
Position-time graph for motion with (a) positive acceleration; (b) negative acceleration, and (c) zero acceleration . Let us see how velocity-time graph looks like for some simple cases. Fig. .
shows velocity- time graph for motion with constant acceleration for the following cases : (a) An object