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Chapter 2: Problem 73

You're an investigator for the National Transportation Safety Board, examining a subway accident in which a train going at \(80 \mathrm{km} / \mathrm{h}\) collided with a slower train traveling in the same direction at \(25 \mathrm{km} / \mathrm{h}\). Your job is to determine the relative speed of the collision, to help establish new crash standards. The faster train's "black box" shows that it began negatively accelerating at \(2.1 \mathrm{m} / \mathrm{s}^{2}\) when it was \(50 \mathrm{m}\) from the slower train, while the slower train continued at constant speed. What do you report?

Short Answer

Expert verified
The relative speed of the two trains at the point of their collision can be computed by following the steps above. This involves solving an equation of motion for time, calculating the final speed of the faster train using that time, and finding the relative speed by comparing this result with the constant speed of the slower train.

Step by step solution

01

Calculate Time

The initial velocity \(v_{0}\) of the faster train is \(80 \mathrm{km} / \mathrm{h}\) or \(22.22 \mathrm{m} / \mathrm{s}\). The acceleration \(a\) is \(-2.1 \mathrm{m} / \mathrm{s}^{2}\) (negative as the train is decelerating). The distance \(s\) it has to cover is 50 m. Implementing these values into the equation of motion \( s = v_{0}*t + \frac{1}{2}*a*t^{2}\), we can solve for \(t\).
02

Solve the Quadratic Equation

In solving for \(t\) we have a quadratic equation with \( t_{1,2} = \frac{-v_{0} \pm \sqrt{v_{0}^{2} - 2 * a * s}}{a} \( and we're interested in the positive root.
03

Find Final Velocity of Faster Train

After finding the time, we can then use it to find the final velocity of the faster train at the point of collision, using the formula \( v = v_{0} + a*t \).
04

Find Relative Speed

The speed of the slower train is \(25 \mathrm{km} / \mathrm{h}\) or \(6.94 \mathrm{m} / \mathrm{s}\). Subtract this speed from the final velocity of the faster train to get the relative speed.

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Most popular questions from this chapter

An object's position as a function of time \(t\) is given by \(x=b t^{4}\) with \(b\) a constant. Find an expression for the instantaneous velocity, and show that the average velocity over the interval from \(t=0\) to any time \(t\) is one- fourth of the instantaneous velocity at \(t\)

A car moving initially at \(50 \mathrm{mi} / \mathrm{h}\) begins slowing at a constant rate \(100 \mathrm{ft}\) short of a stoplight. If the car comes to a full stop just at the light, what is the magnitude of its acceleration?

Taking Earth's orbit to be a circle of radius \(1.5 \times 10^{8} \mathrm{km},\) determine Earth's orbital speed in (a) meters per second and (b) miles per second.

Amtrak's 20 th-Century limited is en route from Chicago to New York at \(110 \mathrm{km} / \mathrm{h}\) when the engineer spots a cow on the track. The train brakes to a halt in 1.2 min, stopping just in front of the cow. (a) What is the magnitude of the train's acceleration? (b) What's the direction of the acceleration? (c) How far was the train from the cow when the engineer applied the brakes?

A castle's defenders throw rocks down on their attackers from a 15-m-high wall, with initial speed \(10 \mathrm{m} / \mathrm{s}\). How much faster are the rocks moving when they hit the ground than if they were simply dropped?
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