As two trains move along a track, their conductors suddenly notice that they are headed toward each other. The figure gives their velocities v as functions of time t as the conductors slow the trains. The figure's vertical scaling is set by vs = 40.0 m/s. The slowing processes begin when the trains are 240 m apart. What is their separation when both trains have stopped?
To find the separation when both trains have stopped, we need to analyze the given velocities and determine when both trains come to a complete stop.
Let's refer to the velocities of the two trains as v1(t) and v2(t), where v1(t) is the velocity of the first train and v2(t) is the velocity of the second train at time t.
Given that the vertical scaling of the figure is set by vs = 40.0 m/s, we can determine the velocities of the two trains at specific times from the figure.
Let's assume v1(t) and v2(t) are points on the figure, where v1(t1) = v1 and v2(t2) = v2:
1. From the figure, determine the velocity v1 at time t1.
2. From the figure, determine the velocity v2 at time t2.
3. Use the given information that the trains start 240 m apart to determine the time it takes for the two trains to meet. We can calculate the time t_meet using the formula:
t_meet = 240 m / (v1(t1) + v2(t2))
4. Now, we need to find the time when both trains come to a complete stop. Let's assume it is t_stop.
To find t_stop, we need to equate v1(t_stop) = 0 and v2(t_stop) = 0. This will give us the time at which both trains have stopped.
5. Finally, substitute the value of t_stop into the equation for the separation of the trains to find their separation when both trains have stopped:
separation = v1(t_stop) * t_stop + v2(t_stop) * t_stop
By following these steps, you can determine the separation when both trains have stopped.