A particle moves along a straight line for seconds
MATHEMATICS · Multi-part · AP · Calculus AB · Applications of Integration · Applications of Integration
Problem
A particle moves along a straight line for 0≤t≤10 seconds. The velocity v(t) (in m/s) is described verbally as follows:
• From t=0 to t=4 the velocity is positive, and the region bounded by the graph of v(t) and the t–axis on this interval has greater area than the corresponding region on 4<t≤10.
• From t=4 to t=10 the velocity is negative.
Assume v(t) is continuous and piecewise smooth, and no other sign changes occur.
Throughout, interpret all answers in the context of the motion of the particle.
(a) Describe how the sign of v(t) on each sub-interval influences the particle’s displacement on [0,10]. Explain without performing any numerical integration.
(b) Explain why the particle’s total distance traveled on [0,10] must exceed the magnitude of its net displacement.
(c) Suppose the area of the region above the t–axis on [0,4] is 26 m and the area of the region below the t–axis on [4,10] is 34 m. Interpret the physical meaning of each of these two numbers and determine whether the particle ends to the right or to the left of its starting position.