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 \(\displaystyle t_1 \lt t \leq t_2 \)?