/*! 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 12 An automobile having a mass of \... [FREE SOLUTION] | 91影视

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Chapter 13: Problem 12

An automobile having a mass of \(1000 \mathrm{~kg}\) is driven into a brick wall in a safety test. The bumper behaves like a spring with constant \(5.00 \times 10^{6} \mathrm{~N} / \mathrm{m}\) and is compressed \(3.16 \mathrm{~cm}\) as the car is brought to rest. What was the speed of the car before impact, assuming no energy is lost in the collision with the wall?

Short Answer

Expert verified
The speed of the car before impact is approximately \(13.95 \mathrm{m/s}\).

Step by step solution

01

Formulate the problem

We're asked to find the speed of the car before the impact, and we know from the conservation of energy that the kinetic energy of the car before the collision (\(\frac{1}{2} mv^{2}\)) is equal to the potential energy stored in the spring (bumper) when it is fully compressed (\(\frac{1}{2} kx^{2}\)), where m is the mass of the car, v is the speed of the car, k is the spring constant and x is the displacement of the spring.
02

Convert the compression of the spring into meters

The compression of the spring, \(x = 3.16 \mathrm{cm} \) is given in centimeters, but the spring constant \(k\) is given in \(N / m\). To ensure consistent units, we convert \(x\) from cm to m by multiplying by 0.01: \(x = 3.16 \mathrm{cm} * 0.01 = 0.0316 \mathrm{m}\).
03

Use conservation of energy to find the speed of the car

Setting the kinetic energy before collision equal to the potential energy after the collision (when the spring is fully compressed) allows us to compute the velocity: \[\frac{1}{2} mv^{2} = \frac{1}{2} kx^{2}\]\[v = \sqrt{\frac{kx^{2}}{m}}\]Substitute the given values into the formula: \[v = \sqrt{\frac{5.00 \times 10^{6}\mathrm{N/m} * (0.0316\mathrm{m})^{2}}{1000\mathrm{kg}}}\]

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Most popular questions from this chapter

A simple pendulum is \(5.00 \mathrm{~m}\) long. (a) What is the period of simple harmonic motion for this pendulum if it is located in an elevator accelerating upward at \(5.00 \mathrm{~m} / \mathrm{s}^{2}\) ? (b) What is its period if the elevator is accelerating downward at \(5.00 \mathrm{~m} / \mathrm{s}^{2}\) ? (c) What is the period of simple harmonic motion for the pendulum if it is placed in a truck that is accelerating horizontally at \(5.00 \mathrm{~m} / \mathrm{s}^{2}\) ?

An ethernet cable is \(4.00 \mathrm{~m}\) long and has a mass of \(0.200 \mathrm{~kg}\). A transverse wave pulse is produced by plucking one end of the taut cable. The pulse makes four trips down and back along the cable in \(0.800 \mathrm{~s}\). What is the tension in the cable?

A harmonic wave is traveling along a rope. It is observed that the oscillator that generates the wave completes \(40.0\) vibrations in \(30.0 \mathrm{~s}\). Also, a given maximum travels \(425 \mathrm{~cm}\) along the rope in \(10.0 \mathrm{~s}\). What is the wavelength?

A horizontal block-spring system with the block on a frictionless surface has total mechanical energy \(E=\) \(47.0 \mathrm{~J}\) and a maximum displacement from equilibrium of \(0.240 \mathrm{~m}\). (a) What is the spring constant? (b) What is the kinetic energy of the system at the equilibrium point? (c) If the maximum speed of the block is \(3.45 \mathrm{~m} / \mathrm{s}\), what is its mass? (d) What is the speed of the block when its displacement is \(0.160 \mathrm{~m}\) ? (e) Find the kinetic energy of the block at \(x=0.160 \mathrm{~m}\). (f) Find the potential energy stored in the spring when \(x=0.160 \mathrm{~m}\). (g) Suppose the same system is released from rest at \(x=0.240 \mathrm{~m}\) on a rough surface so that it loses \(14.0 \mathrm{~J}\) by the time it reaches its first turning point (after passing equilibrium at \(x=0\) ). What is its position at that instant?

The position of an object connected to a spring varies with time according to the expression \(x=\) \((5.2 \mathrm{~cm}) \sin (8.0 \pi t)\). Find (a) the period of this motion, (b) the frequency of the motion, (c) the amplitude of the motion, and (d) the first time after \(t=0\) that the object reaches the position \(x=2.6 \mathrm{~cm}\).
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