As two trains move along a track, their conductors suddenly notice that they are headed toward each other. The figure below gives their velocities v as functions of time t as the conductors slow the trains. The slowing processes begin when the trains are 240 m apart. What is their separation when both trains have stopped?

Question

As two trains move along a track, their conductors suddenly notice that they are headed toward each other. The figure below gives their velocities v as functions of time t as the conductors slow the trains. The slowing processes begin when the trains are 240 m apart. What is their separation when both trains have stopped?

Do I use v= v_0+at or x-x_0=(v_0)(t)+0.5at^2 to solve this?

Answer 1

To solve this problem, we need to determine the separation between the trains when they have both stopped. Since the velocities of the trains are given as functions of time, we can use the equations of motion to find the separation at the final time when both trains have come to a stop.

Let's break this problem down into steps:

Step 1: Analyze the given information
According to the problem, the velocities of the trains are given as functions of time, v(t). We are also given that the slowing processes begin when the trains are 240 m apart.

Step 2: Determine the stopping times for both trains
To find the separation when both trains have stopped, we need to determine the time at which each train comes to a stop. We can determine this by finding the time at which the velocity of each train becomes zero.

Step 3: Calculate the separation at the final time
Once we have the stopping times for both trains, we can substitute these values into the equation for separation to find the final separation.

Now, let's address your question about which equation to use. In this case, we will use the equation:

x - x_0 = (v_0)(t) + (1/2)(a)(t^2)

This equation relates the separation distance (x), initial separation distance (x_0), initial velocity (v_0), acceleration (a), and time (t).

Step 4: Solve the problem
To solve the problem, follow these steps:

1. Start with the separation equation:
x - x_0 = (v_0)(t) + (1/2)(a)(t^2)

2. Substitute the known values:
x - 240 = 0 + (1/2)(a)(t^2)

3. Determine the stopping time for each train:
Set the velocity equation for each train equal to zero and solve for t:

For Train 1: v_1(t) = -20 + 5t = 0
Solve for t_1: -20 + 5t_1 = 0

For Train 2: v_2(t) = -15 + 4t = 0
Solve for t_2: -15 + 4t_2 = 0

4. Substitute the stopping times into the equation for separation:
x - 240 = 0 + (1/2)(a)(t^2)

For Train 1: x - 240 = 0 + (1/2)(a)(t_1^2)
For Train 2: x - 240 = 0 + (1/2)(a)(t_2^2)

5. Solve the two equations simultaneously to find the separation at the final time.

By following these steps and using the equation for separation, you can find the final separation between the trains when they both come to a stop.