Pageiot? ....... Explore Acceleration The difference in length between two successive arrows (velocity vectors) on a motion diagram indicates the relative magnitude and direction of the change in velocity of the object over time. This new arrow is the velocity change arrow and has the same direction. though not the same length. as the acceleration vector. illustrated in Figure 5. Acceleration is the time rate of change of the velocity and is a vector quantity rneasu red in units of (mr's),"s or Wit/52.1116 equation for average acceleration can be written as: where a' average acceleration. v17 Vi] : change in velocity. and tlr t0 : elapsed time. a At = O sec t = 2 sec t = 4 sec Positive acceleration, speeding up a B. t = 4 sec t = 2 sec t = 0 sec Positive acceleration, slowing down c. t= 0 sec t = 2 sec t = 4 sec Negative acceleration, slowing down t = 4 sec t = 2 sec t = 0 sec D. .+ Negative acceleration, speeding up Figure 5. Motion diagrams with acceleration vectors. A. This diagram depicts positive acceleration and velocity. B. This diagram depicts positive acceleration and negative velocity. C. This diagram depicts negative acceleration and positive velocity. D. This diagram depicts negative acceleration and velocity. Instantaneous acceleration is found by reducing the elapsed time in the above equation to a very small value. Using calculus, the acceleration is the derivative of the velocity with respect to time:Figure 5. Motion diagrams with acceleration vectors. A. This diagram depicts positive acceleration and velocity. B. This diagram depicts positive acceleration and negative velocity. C. This diagram depicts negative acceleration and positive velocity. D. This diagram depicts negative acceleration and velocity. instantaneous acceleration is found by reducing the elapsed ti me in the above equation to a very small value. Using calculus. the acceleration is the derivative of the velocity with respect to ti me: where a = instantaneous acceleration and dw'dt is the time derivative of the velocity. The acceleration can also be represented bythe second derivative ofthe displacement with respect to time: where a\" = instantaneous acceleration and dzxr'dt2 = the second time derivative of the xcomponent of the displacement. When the acceleration vector points in the same direction as the velocity vector. the velocity increases. By contrast. when the acceleration vector points in the opposite direction as the velocity vector. the velocity decreases. This second case is often referred to as deceleration. The sign of the acceleration can be either positive or negative for either case. The important information is the sign of the acceleration in conjunction with the sign of the velocity. as depicted in Figure 5, above