Villaretq0

2022-06-24

Antiderivatives: Car Deceleration Problem
A car braked with a constant deceleration of , producing skid marks measuring 160 ft before coming to a stop. How fast was the car travelling when the brakes were first applied?
I know I can solve this problem using kinematics equations from physics; using ${v}_{f}^{2}={v}_{i}^{2}+2ad$ yields an initial velocity of 113 ft/s. However, I am supposed to be using antiderivatives and not physics. So far, I figured that if $a\left(t\right)=-40$ then $v\left(t\right)=-40t+{c}_{1}$ and $d\left(t\right)=-20{t}^{2}+{c}_{1}x+{c}_{2}$, where ${c}_{1}$ and ${c}_{2}$ are constants. I'm not quite sure what my next step should be... any suggestions?

popman14ee

Expert

Explanation:
Your equations are correct ( with a t instead of x in the second equation) and, if we measure the time and the space from the instant in which began the braking, in your second equation we have ${c}_{2}=0$ ( the initial position is $d\left(0\right)=0$). the other constant ${c}_{1}$ is the inital velocity $v\left(0\right)={c}_{1}$ and at the instant ${t}_{1}$ in wich the car stops its motion we have:
$\left\{\begin{array}{l}0=-40{t}_{1}+v\left(0\right)\\ 160=-20{t}_{1}^{2}+v\left(0\right){t}_{1}\end{array}$ solve this system and you find the duration of the braking ${t}_{1}$ and the inital velocity v(0).

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