/*! 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 48 The race car starts from rest an... [FREE SOLUTION] | 91影视

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Chapter 12: Problem 48

The race car starts from rest and travels along a straight road until it reaches a speed of \(26 \mathrm{~m} / \mathrm{s}\) in \(8 \mathrm{~s}\) as shown on the \(v-t\) graph. The flat part of the graph is caused by shifting gears. Draw the \(a-t\) graph and determine the maximum acceleration of the car.

Short Answer

Expert verified
The maximum acceleration of the race car is \( 3.25 \mathrm{~m/s^2} \)

Step by step solution

01

Understand the v-t graph

The given graph has an incline in the first part till reaching the speed of \(26 \mathrm{~m/s}\), a flat part representing the gear shift where velocity is constant, and a final incline. Respectively, the acceleration would be present in the incline portions and zero where the graph is flat because acceleration is the derivative (slope) of the velocity versus time graph.
02

Calculate the Acceleration

The acceleration during the incline part can be calculated by the formula \( a=\Delta v/\Delta t \), where \( \Delta v \) is the change in velocity and \( \Delta t \) is the change in time. In the given exercise, the velocity changes from \(0 \mathrm{~m/s}\) to \(26 \mathrm{~m/s}\) in a span of \(8 \mathrm{~s}\) before the gear shift. So, acceleration, \( a \) is \( a = (26\mathrm{~m/s}- 0 \mathrm{~m/s})/(8\mathrm{~s}) \)
03

Determine the Maximum Acceleration

Based on the calculation in previous step, the maximum acceleration of the car occurs during the incline part of the \( v-t \) graph and it is calculated as \( a = 3.25 \mathrm{~m/s^2} \). Once the car starts shifting gears, acceleration drops to zero since the velocity is constant.
04

Draw the a-t Graph

To draw the \( a-t \) graph, one would start by plotting the positive acceleration of \( 3.25 \mathrm{~m/s^2} \) for the first \(8\mathrm{~s}\). This will be a straight line parallel to the x-axis since the acceleration is constant. Then, plot a horizontal line on the x-axis representing zero acceleration for the remaining time during which the car is shifting gears. For the final incline, assuming it has the same slope as the initial part, the acceleration would again be \( 3.25 \mathrm{~m/s^2} \).

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Most popular questions from this chapter

A boat is traveling along a circular curve having a radius of \(100 \mathrm{ft}\). If its speed at \(t=0\) is \(15 \mathrm{ft} / \mathrm{s}\) and is increasing at \(\dot{v}=(0.8 t) \mathrm{ft} / \mathrm{s}^{2}\), determine the magnitude of its acceleration at the instant \(t=5 \mathrm{~s}\).

If the position of a particle is defined by \(s=[2 \sin (\pi / 5) t+4] \mathrm{m}\), where \(t\) is in seconds, construct the \(s-t, v-t\), and \(a-t\) graphs for \(0 \leq t \leq 10 \mathrm{~s}\).

A horse on the merry-go-round moves according to the equations \(r=8 \mathrm{ft}, \theta=(0.6 t) \mathrm{rad}\), and \(z=(1.5 \sin \theta) \mathrm{ft}\), where \(t\) is in seconds. Determine the cylindrical components of the velocity and acceleration of the horse when \(t=4 \mathrm{~s}\).

The particle travels along the path defined by the parabola \(y=0.5 x^{2} .\) If the component of velocity along the \(x\) axis is \(v_{x}=(5 t) \mathrm{ft} / \mathrm{s}\), where \(t\) is in seconds, determine the particle's distance from the origin \(O\) and the magnitude of its acceleration when \(t=1 \mathrm{~s} .\) When \(t=0, x=0, y=0 .\)

Show that the girl at \(A\) can throw the ball to the boy at \(B\) by launching it at equal angles measured up or down from a \(45^{\circ}\) inclination. If \(v_{A}=10 \mathrm{~m} / \mathrm{s}\), determine the range \(R\) if this value is \(15^{\circ}\), i.e., \(\theta_{1}=45^{\circ}-15^{\circ}=30^{\circ}\) and \(\theta_{2}=45^{\circ}+15^{\circ}=60^{\circ}\). Assume the ball is caught at the same elevation from which it is thrown.
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