/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 44 A train \(400 \mathrm{~m}\) long... [FREE SOLUTION] | 91影视

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Chapter 2: Problem 44

A train \(400 \mathrm{~m}\) long is moving on a straight track with a speed of \(82.4 \mathrm{~km} / \mathrm{h}\). The engineer applies the brakes at a crossing, and later the last car passes the crossing with a speed of \(16.4 \mathrm{~km} / \mathrm{h}\). Assuming constant acceleration, determine how long the train blocked the crossing. Disregard the width of the crossing.

Short Answer

Expert verified
The train blocks the crossing for approximately 29.18 seconds.

Step by step solution

01

Conversion of units

First, let's convert all the speeds from \( \mathrm{~km/h} \) to \( \mathrm{~m/s} \) to keep the units consistent. 1 kilometer per hour (km/h) is approximately equal to 0.27778 meters per second (m/s). This gives us: Initial velocity \( v_i = 82.4~ \mathrm{km/h} = 22.88 ~\mathrm{m/s} \) and Final velocity \( v_f = 16.4~ \mathrm{km/h} = 4.56 ~\mathrm{m/s}\).
02

Use the third equation of motion

The time taken to clear the crossing is equal to the time needed for the train to decelerate from its initial velocity to its final velocity. According to the third equation of motion, \( v_f = v_i + a\cdot t \), which can rearranged to find time \( t = (v_f - v_i) / a \). But in this case, we don't have the value for acceleration \( a \).
03

Use the first equation of motion to find acceleration

We can use the first equation of motion \( v_f = v_i + 2a\cdot s \) where \( s \) is the distance which is the length of the train (400 m). Rearranging this gives us: \( a = (v_f^2 - v_i^2) / (2s) = (4.56^2 - 22.88^2) / (2 \cdot 400~\mathrm{m}) = -0.6286 ~\mathrm{m/s^2} \). The negative sign indicates that it's a deceleration.
04

Substitute the value of acceleration into the time equation

Now let's substitute the value of \( a \) into the time equation to get the time: \( t = (v_f - v_i) / a = (4.56~\mathrm{m/s} - 22.88~\mathrm{m/s}) / -0.6286~\mathrm{m/s^2} = 29.175~\mathrm{s} \).

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