A particle is released from rest at the origin at time seconds and moves along a…
MATHEMATICS · Multi-part · AP · Calculus AB · Contextual Applications of Differentiation · Contextual Applications of Differentiation
Problem
A particle A is released from rest at the origin at time t=0 seconds and moves along a straight track with constant acceleration aA=2 m/s2 directed toward the positive x–axis. Four seconds later, a second particle B is projected from x=30 m toward the origin with an initial speed of 12 m/s. After launch, particle B experiences a time–dependent acceleration aB(t)=1.5−0.25t (in m/s2) for t≥4, where t denotes the time measured from t=0. Assume both particles stay on the line and do not change direction once launched.
(a) For 0≤t≤4, write explicit formulas for the velocity and position of particle A.
(b) For t≥4, derive expressions for the velocity vB(t) and position sB(t) of particle B measured from the origin.
(c) Obtain an expression for the separation distance D(t)=sB(t)−sA(t) for t≥4.
(d) Determine the first time t>4 when the particles meet. Justify that this is the only meeting time.
(e) Compute the velocity of each particle at the meeting time and interpret the sign of each velocity in the context of the scenario.