Will the Train Stop in Time?
Date: 09/21/1999 at 17:57:48 From: Student Subject: Train/speed/brakes A train is traveling at 80.0 m/s when the engineer sees a car stalled on the tracks 2000 m ahead. He immediately applies the brakes and the train starts to slow down at the rate of 1.50 m/s^2. Will the train stop in time? If so, how far short does it stop? If not, with what speed does the train hit the car, and how far past the car does the train go?
Date: 10/08/1999 at 13:24:58 From: Doctor TWE Subject: Re: Train/speed/brakes Hi - There are four formulae in physics that can be useful in solving this type of problem. They are: (1) s = .5(u + v)t (2) v = u + at (3) s = ut + .5a(t^2) (4) v^2 = u^2 + 2as where: s = distance traveled u = initial velocity (speed) t = time a = acceleration v = final velocity (speed) Note that the acceleration will be negative because the train is decelerating (slowing down). Let's first look at what we have, and what we need to know. We know the initial velocity of the train, u = 80 m/s. We also know that the acceleration a = -1.5 m/s^2. We don't know the stopping distance, s, (we want to see if it is less than 2000 m); nor do we know the time it takes to stop, t. If the train comes to a complete stop, we know the final velocity, v = 0. We can start by finding the stopping distance (s) and see whether it is more or less than 2000 m. Both equation (1) and equation (3) solve for s, but to use either one we need to first find the stopping time, t. To find the time, we need to look at the equations with t. The only equation other than (1) and (3) that involves t is equation (2). Can we use it? We know a, u and v (once the train stops, the final velocity is zero), so we can use this to find t. Can you rearrange the equation to solve for t? If not, write back and I'll explain how to rearrange it. Once we have the stopping time t, we can find the stopping distance using either equation (1) or (3). Which one should we use? How about trying both and seeing if you get the same answer. If you do - great; it's probably right. If you get two different answers, check your arithmetic and double-check your steps in finding the time. Now we know whether the train stopped in time. If it did, we can also compute how far short of the car it stopped. (Remember that s was the stopping distance of the train, not how far short of the car it stopped.) If not, you want to find the speed at the time of impact and how far past the car the train traveled. If the train hits the car, we need to calculate the impact velocity v. Lets re-evaluate what we know in this case. We know the initial velocity u, the acceleration a, and the distance s. The time we calculated above was to come to a complete stop, so it is not relevant if the train actually hits the car. Equations (1), (2) and (4) involve v, but (1) and (2) require t to find v. So we have to use equation (4). Since the left side of equation (4) is v^2, remember to take the square root after computing the stuff on the right side. (We want to find v, not v^2.) How far the train travels after hitting the car depends in part on the effect the car has on the rate of the train's deceleration (the car will further "slow down" the train when they hit). For the sake of the problem, let's assume that the car's effect is negligible. This is not too unrealistic - the train has a mass many times that of the car, and therefore the car will not significantly affect it. (Imagine a bowling ball hitting a ping pong ball.) We can find the total distance the train travels using the method above, then compute how far past the car that is. To see another similar problem, check out the following problem in our Ask Dr. Math archives: http://mathforum.org/dr.math/problems/mangan4.9.97.html I hope this helps. Write back if you have any more questions. - Doctor TWE, The Math Forum http://mathforum.org/dr.math/
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