A particle’s velocity (in ft/s) is modeled for seconds by two geometric regions: • For
MATHEMATICS · Multi-part · AP · Calculus AB · Applications of Integration · Applications of Integration
Problem
A particle’s velocity v(t) (in ft/s) is modeled for 0≤t≤10 seconds by two geometric regions:
• For 0≤t≤a, v(t) is constant at 5.
• For a<t≤10, v(t) decreases linearly from 5 at t=a to −3 at t=10.
The net displacement of the particle over the entire 10-second interval is known to be 12 feet.
(a) Express the particle’s displacement on [0,a] and on [a,10] in terms of a, using geometric areas.
(b) Form an equation in a whose solution ensures the net displacement is 12 feet, and solve for a.
(c) Compute the total distance traveled on [0,10] using the value of a found in part (b).
(d) Discuss how varying a (while holding the endpoints of the velocity graph fixed) would affect the net displacement and total distance. Support your discussion with reasoning based on areas under the velocity curve.