College physics question?????

Question

Two trains, one traveling at 60.00 km/h and the other at 120.00 km/h, are headed toward one another along a straight, level track. When they are 629 m apart, each engineer sees the other's train and applies the brakes. The brakes decelerate each train at the rate of 1.0 m/s2. Is there a collision?

What distance do the trains need to allow between them to stop at this acceleration?

What acceleration do the two trains need to have to stop exactly in a distance of 629 m?

Answer 1

You should know the displacement equation, which is x = v0t + 1/2at^2. Find the derivative with respect to t (dx/dt). You get the equation for final velocity, which is v = v0 + at.  So you want to know when each stops right? And they stop when their final velocity is 0. So you plug in the initial velocities and deceleration rates of each train into the final velocity equation (which is 0).  Converting units properly, we

get for the 60 km/h train:

0 = 16 2/3 m/s - 1 m/s^2 (t) ----->time it takes to stop = 16 2/3 seconds

for 120 km/h train

0 = 33 1/3 - 1 m/s^2 (t) -----> time it takes to stop = 33 1/3 seconds

Then plug in times into the displacement equation:

for 60km/h train: x = 16 2/3(16 2/3) - 1 m/s^2*(1/2)*(16 2/3)^2 =

travels 138 8/9 meters until it stops

for 120km/h train: x = 33 1/3(33 1/3) - 1 m/s^2*(1/2)*(33 1/3)^2 =

travels 555.56 meters until it stops

----->Adding the two distances: 555.56 + 138 8/9 = about 694.44 meters, which exceeds the distance they notice each other. So, they collide.