AP Calculus · free response
Particle Motion
No calculator · 8 points
A particle moves along a line with velocity v(t)=t-4 (meters per second) for 0\le t\le 5.
(a) Find the acceleration a(4) of the particle.
a(t)=v′(t)=1
(b) Find the displacement of the particle over 0\le t\le 5.
Displacement is \int_0^{5} v(t)\,dt
Evaluate the displacement integral
Displacement is in meters
(c) Find the total distance traveled over 0\le t\le 5.
Total distance is \int_0^{5} |v(t)|\,dt (split at t=4, where v=0)
Evaluate the distance integral
(d) For what times t is the particle moving left? Justify.
Particle moves left where v(t)<0, i.e. 0\le t<4; justified by the sign of v
(e) Using correct units, interpret the meaning of \int_0^{5} v(t)\,dt in the context of the particle's motion.
It is the net change in position (displacement) over [0,5], in meters