# PS.1.3 Worked Example: Braking Car

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At time $$\displaystyle t=0$$, a car moving along the +$$\displaystyle x$$-axis passes through $$\displaystyle x=0$$ with a constant velocity of magnitude $$\displaystyle v_0$$. At some time later, $$\displaystyle t_1$$, it starts to slow down. The acceleration of the car as a function of time is given by:

 $$\displaystyle a(t) = \left\{ \begin{array}{ll} 0 & \quad 0 \leq t \leq t_1 \\ -c(t-t_1) & \quad t_1 \lt t \leq t_2 \end{array} \right.$$

where $$\displaystyle c$$ is a positive constants in SI units, and $$\displaystyle t_1 \lt t \leq t_2$$ is the given time interval for which the car is slowing down. The goal of the problem is to find the car's position as a function of time between $$\displaystyle t_{1} \lt t \lt t_2$$. Express your answer in terms of v_0 for $$\displaystyle v_0$$, t_1 for $$\displaystyle t_1$$, t_2 for $$\displaystyle t_2$$, and $$\displaystyle c$$ as needed.

(Part a). What is $$\displaystyle v(t)$$, the velocity of the car as a function of time during the time interval $$\displaystyle 0 \leq t \leq t_1$$?

(Part b). What is $$\displaystyle x(t)$$, the position of the car as a function of time during the time interval $$\displaystyle 0 \leq t \leq t_1$$?

(Part c). What is $$\displaystyle v(t)$$, the velocity of the car as a function of time during the time interval $$\displaystyle t_1 \lt t \leq t_2$$?

(Part d). What is $$\displaystyle x(t)$$, the position of the car as a function of time during the time interval $$\displaystyle t_1 \lt t \leq t_2$$?

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