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Chapter 2: Problem 83

A train traveling at \(40.0 \mathrm{~m} / \mathrm{s}\) is headed straight toward another train, which is at rest on the same track. The moving train decelerates at \(6.0 \mathrm{~m} / \mathrm{s}^{2},\) and the stationary train is \(100.0 \mathrm{~m}\) away. How far from the stationary train will the moving train be when it comes to a stop?

Short Answer

Expert verified
Answer: The moving train will be approximately 33.3 meters away from the stationary train when it comes to a stop.

Step by step solution

01

Find time taken for the moving train to stop

To find out the time it takes for the moving train to come to a stop, we will use the following equation of motion: \(v = u + at\) Where: \(v\) - final velocity (0 m/s since the train comes to a stop) \(u\) - initial velocity (40.0 m/s, given) \(a\) - acceleration (-6.0 m/s², since it is decelerating) \(t\) - time (the unknown we are solving for) Now, let's plug in the given values into the equation and solve for the time `t`. \(0 = 40.0 + (-6.0) \cdot t\)
02

Solve for time 't'

Solving for t in the equation: \(-40.0 = -6.0t\) \(t = \frac{-40.0}{-6.0}\) \(t = 6.\bar{6} \ \text{seconds}\)
03

Calculate the distance traveled by the moving train

Now, we will use another equation of motion to find the distance covered by the moving train during the time it took to come to a stop. \(s = ut + \frac{1}{2}at^2\) Where: \(s\) - distance traveled (unknown we are solving for) \(u\) - initial velocity (40.0 m/s, given) \(a\) - acceleration (-6.0 m/s², since it is decelerating) \(t\) - time (6.\bar{6} seconds, found in step 2) Now let's plug in the values into the equation and solve for the distance `s`. \(s = 40.0 \cdot 6.\bar{6} + \frac{1}{2} \cdot (-6.0) \cdot (6.\bar{6})^2\)
04

Solve for distance 's'

Solving for s in the equation: \(s = 40.0 \cdot 6.\bar{6} - 3.0 \cdot (6.\bar{6})^2\) \(s \approx 133.3 \ \text{meters}\)
05

Calculate the distance between the stationary train and the moving train when it stops

We now know that the moving train traveled approximately 133.3 meters before coming to a stop. Since the initial distance between the trains was 100 meters, we need to calculate the difference to find out how far the moving train is from the stationary train when it stops. Distance between the trains when moving train stops = Distance traveled by moving train - Initial distance between trains \(\approx 133.3 - 100.0 = 33.3 \ \text{meters}\) So, the moving train will be approximately 33.3 meters away from the stationary train when it comes to a stop.

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Most popular questions from this chapter

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