Car R travels eastward along a straight highway and passes the coordinate origin at…
MATHEMATICS · Multi-part · AP · Calculus AB · Contextual Applications of Differentiation · Contextual Applications of Differentiation
Problem
Car R travels eastward along a straight highway and passes the coordinate origin O at time t=0 seconds with a speed of 8 m/s. Its acceleration for t≥0 is given by aR(t)=6−2t (in m/s2). Car S is located 260 m east of O at t=0 and is moving westward toward O with velocity vS(t)=−(10+0.5t) (in m/s). Positive position is measured eastward.
(a) Derive expressions for the velocity vR(t) and position xR(t) of Car R for t≥0.
(b) Verify that Car S maintains westward motion for all t≥0 by analyzing vS(t).
(c) Formulate an equation whose solution(s) give(s) the collision time(s) of the two cars. Clearly state the physical principle used to set up this equation.
(d) Solve the equation from part (c) to determine whether the cars collide. If they do, state the collision time and location; if they do not, explain why the collision cannot occur.
(e) At the collision time found in part (d) (if any), compute each car’s speed and indicate which car experiences greater magnitude of deceleration at that instant.