A particle moves along a straight line with velocity
MATHEMATICS · Multi-part · AP · Calculus AB · Applications of Integration · Applications of Integration
Problem
A particle moves along a straight line with velocity v(t), measured in meters per second, for 0≤t≤5 seconds.
• For 0≤t<3, the velocity is positive and decreases continuously from 6 m/s at t=0 to 0 m/s at t=3.
• For 3<t≤5, the velocity is negative and its magnitude increases continuously from 0 m/s just after t=3 to −4 m/s at t=5.
Use this information to analyze the motion of the particle over the entire five–second interval.
(a) Describe the conceptual difference between the particle’s net displacement and its total distance traveled on the interval 0≤t≤5. Include a discussion of sign, orientation, and the role of the velocity function in each quantity.
(b) Without performing numerical integration, explain how the areas between the velocity graph and the t–axis over the sub-intervals [0,3] and [3,5] can be combined to obtain (i) the net displacement and (ii) the total distance traveled.
(c) Decide which quantity—net displacement or total distance traveled—must be larger in magnitude for this motion. Provide a clear justification based on your reasoning from part (b).