/*! 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 39 Two cars, one a compact with mas... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 8: Problem 39

Two cars, one a compact with mass 1200 \(\mathrm{kg}\) and the other a large gas-guzzler with mass \(3000 \mathrm{kg},\) collide head-on at typical freeway speeds. (a) Which car has a greater magnitude of momentum change? Which car has a greater velocity change? (b) If the larger car changes its velocity by \(\Delta v\) , calculate the change in the velocity of the small car in terms of \(\Delta v .\) (c) Which car's occupants would you expect to sustain greater injuries? Explain.

Short Answer

Expert verified
Both cars have equal momentum change; compact car has greater velocity change. The compact car's velocity change is 2.5 times greater. Compact car's occupants likely sustain greater injuries.

Step by step solution

01

Momentum Change Analysis

In a head-on collision, according to the law of conservation of momentum, the total momentum before collision equals the total momentum after collision. During the collision, the momentum lost by one car is gained by the other. Therefore, both cars experience the same magnitude of momentum change, even though their masses are different.
02

Velocity Change Analysis

The change in velocity is given by the formula \( \Delta v = \frac{\Delta p}{m} \), where \( \Delta p \) is the change in momentum and \( m \) is the mass. Since momentum change \( \Delta p \) is the same for both cars, the smaller mass (the compact car) will result in a greater velocity change. Thus, the compact car will have a greater velocity change.
03

Calculate Velocity Change of Small Car

Let the velocity change of the large car be \( \Delta v \). The large car's momentum change is \( m_{large} \cdot \Delta v = 3000\, \mathrm{kg} \times \Delta v \).For the compact car, momentum change is equal to the large car's momentum change, so:\[ m_{small} \cdot \Delta v_{small} = 3000\, \mathrm{kg} \times \Delta v \]Solving for \( \Delta v_{small} \), we get:\[ \Delta v_{small} = \frac{3000}{1200} \times \Delta v = 2.5 \cdot \Delta v \]Therefore, the change in velocity of the compact car is 2.5 times the change in velocity of the large car.
04

Evaluate Occupant Injuries

In a collision, the occupants of the car with greater velocity change experience more rapid deceleration, which typically leads to greater force applied to their bodies (as force is tied to acceleration through Newton's second law \( F = m \cdot a \)). Therefore, the occupants of the compact car, which undergoes a greater velocity change, would likely sustain greater injuries.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Conservation of Momentum
When two objects collide, their total momentum before the collision is the same as their total momentum after the collision. This is known as the conservation of momentum principle.

In the case of our two cars colliding, the compact car and the gas-guzzler, the rule of conservation of momentum tells us that the total momentum they share doesn't change despite the crash.

Here's how it works:
  • The compact car, with a mass of 1200 kg, and the large car, with a mass of 3000 kg, both have momentum dependent on their velocities just before the crash.
  • During the collision, the momentum one car loses equals the momentum the other car gains.
  • So, their momentum changes are equal and opposite, leading to no net momentum change in the system.
By maintaining the overall momentum, we see that while the collisions appear chaotic, the principle of momentum conservation brings an underlying order.
Velocity Change
Velocity change is a crucial part we must consider when discussing collisions. Even though the cars have identical momentum changes due to conservation, their masses influence how much their velocities change.

The formula used here is \( \Delta v = \frac{\Delta p}{m} \), where \( \Delta p \) is the change in momentum and \( m \) is the mass of the car.
  • Both cars undergo the same change in momentum. However, the compact car, because of its lighter mass, experiences a larger velocity change.
  • If the larger car's velocity changes by \( \Delta v \), the formula shows that the compact car鈥檚 velocity changes by 2.5 times that of the large car.
  • This occurs because a smaller mass means a more significant velocity change for a given amount of momentum change.
Thus, velocity change not only reflects the impact of mass but also highlights its effect on occupants' experiences in a collision.
Force and Acceleration
In a collision, understanding force and acceleration is integral to predicting the effects on a vehicle's occupants. When velocity changes, acceleration occurs, and this is what ultimately causes force.

Using Newton's second law, \( F = m \cdot a \), we can comprehend that:
  • The acceleration is directly related to the change in velocity; more dramatic changes in velocity lead to higher accelerations.
  • For the compact car, which undergoes a greater velocity change, the resulting acceleration is higher.
  • This acceleration translates into a larger force experienced by the car's occupants.
Rapid deceleration or a significant reduction in speed means the occupants feel a stronger impact force.

Therefore, even though both cars undergo the same change in momentum, the compact car鈥檚 greater change in velocity and acceleration levels suggests its occupants might sustain more severe injuries. This underscores the importance of vehicle safety features designed to mitigate such forces.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

A small wooden block with mass 0.800 \(\mathrm{kg}\) is suspended from the lower end of a light cord that is 1.60 \(\mathrm{m}\) long. The block is initially at rest. A bullet with mass 12.0 \(\mathrm{g}\) is fired at the block with a horizontal velocity \(v_{0} .\) The bullet strikes the block and becomes embedded in it. After the collision the combined object swings on the end of the cord. When the block has risen a vertical height of \(0.800 \mathrm{m},\) the tension in the cord is 4.80 \(\mathrm{N} .\) What was the initial speed \(v_{0}\) of the bullet?

A 15.0 -kg fish swimming at 1.10 \(\mathrm{m} / \mathrm{s}\) suddenly gobbles up a 4.50 -kg fish that is initially stationary. Neglect any drag effects of the water. (a) Find the speed of the large fish just after it eats the small one. (b) How much mechanical energy was dissipated during this meal?

A \(1050\) -kg sports car is moving westbound at 15.0 \(\mathrm{m} / \mathrm{s}\) on a level road when it collides with a 6320 -kg truck driving east on the same road at 10.0 \(\mathrm{m} / \mathrm{s}\) . The two vehicles remain locked together after the collision. (a) What is the velocity (magnitude and direction) of the two vehicles just after the collision? (b) At what speed should the truck have been moving so that it and the car are both stopped in the collision? (c) Find the change in kinetic energy of the system of two vehicles for the situations of part (a) and part (b). For which situation is the change in kinetic energy greater in magnitude?

A \(68.5-k g\) astronaut is doing a repair in space on the orbit ing space station. She throws a 2.25 -kg tool away from her at 3.20 \(\mathrm{m} / \mathrm{s}\) relative to the space station. With what speed and in what direction will she begin to move?

Accident Analysis. Two cars collide at an intersection. Car \(A,\) with a mass of 2000 \(\mathrm{kg}\) , is going from west to east, while car \(B,\) of mass \(1500 \mathrm{kg},\) is going from north to south at 15 \(\mathrm{m} / \mathrm{s}\) . As a result of this collision, the two cars become enmeshed and move as one afterward. In your role as an expert witness, you inspect the scene and determine that, after the collision, the enmeshed cars moved at an angle of \(65^{\circ}\) south of east from the point of impact. (a) How fast were the enmeshed cars moving just after the collision? (b) How fast was car \(A\) going just before the collision?
See all solutions

Recommended explanations on Physics Textbooks

Fields in Physics

Read Explanation

Physics of Motion

Read Explanation

Electricity

Read Explanation

Kinematics Physics

Read Explanation

Translational Dynamics

Read Explanation

Oscillations

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.