PS.1.5 Worked Example: Pedestrian and Bike at Intersection

« Previous | Next »

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.

Flash and JavaScript are required for this feature.

« Previous | Next »