You are in a car standing by a traffic light and at time \(\displaystyle t=0 \) the light turns green. You start to accelerate during the first \(\displaystyle t_1 \) seconds so that the acceleration of your car is given by:

\(\displaystyle a_1(t) = \left\{ \begin{array}{ll} b_1 & \quad 0 \leq t \leq t_1 \\ 0 & \quad t_1 < t \leq t_2 \end{array} \right. \) |

where \(\displaystyle b_1 \) is a positive constant.

At the instant the light turns green a cyclist passes through the intersection moving with a speed \(\displaystyle v_0 \) in the same direction as your car is moving. At that instant, the cyclist starts to brake with a constant acceleration of magnitude \(\displaystyle b_2 \). At time \(\displaystyle t = t_2 \) the cyclist stops at the same location where you are.

The goal of the problem is to calculate the value of \(\displaystyle b_2 \) in terms of the given variables, \(\displaystyle b_1 \), \(\displaystyle v_0 \), \(\displaystyle t_1 \) and \(\displaystyle t_2 \).

**First:** Describe the motion of the car. Given the car's acceleration \(\displaystyle a_1(t) \), find its velocity and its position as a function of time:

**(Part b)** Calculate \(\displaystyle x_1(t) \) , the car's position as a funcion of time. Express your answer in terms of \(\displaystyle t \), \(\displaystyle t_1 \), and \(\displaystyle b_1 \) as needed.

**Second:** Describe the motion of the bicycle. Given the bicycle's acceleration \(\displaystyle a_2(t) \), find its velocity and its position:

**(Part c)** Calculate \(\displaystyle v_2(t) \) , the bicycle's velocity as a funcion of time. Express your answer in terms of \(\displaystyle t \), \(\displaystyle b_2 \), \(\displaystyle v_0 \) as needed.

**(Part d)** Calculate \(\displaystyle x_2(t) \) , the bicycle's position as a funcion of time. Express your answer in terms of \(\displaystyle t \),\(\displaystyle b_2 \), and \(\displaystyle v_0 \) as needed.

**Third:** Find the value of \(\displaystyle b_2 \). We know that the bicycle stops at \(\displaystyle t=t_2 \), this condition is expressed as:

\(\displaystyle v_2(t_2) = 0 \) (eq. 1)

We also know that the bicycle and the car are at the same location when the bicycle stops. This condition implies:

\(\displaystyle x_2(t_2) = x_1(t_2) \) (eq. 2)

Use (eq.1) and (eq. 2) to obtain the value of \(\displaystyle b_2 \). Express your answer in terms of \(\displaystyle t_1 \), \(\displaystyle b_1 \), and \(\displaystyle v_0 \). Do not use \(\displaystyle t_2 \) in your answer.

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