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Chapter 7: Problem 46

A car (mass \(=1100 \mathrm{~kg}\) ) is traveling at \(32 \mathrm{~m} / \mathrm{s}\) when it collides head-on with a sport utility vehicle (mass \(=2500 \mathrm{~kg}\) ) traveling in the opposite direction. In the collision, the two vehicles come to a halt. At what speed was the sport utility vehicle traveling?

Short Answer

Expert verified
The SUV was traveling at 14.08 m/s.

Step by step solution

01

Understanding the Problem

We are given two vehicles: a car with a mass of 1100 kg moving at 32 m/s, and a sport utility vehicle (SUV) with a mass of 2500 kg moving in the opposite direction. Both vehicles come to a stop after a head-on collision. We need to determine the speed of the SUV before the collision.
02

Apply the Law of Conservation of Momentum

The law of conservation of momentum states that the total momentum before the collision is equal to the total momentum after the collision, provided no external forces act on the system. Here, since the vehicles come to rest after the collision, the final total momentum is 0.
03

Set Up the Initial Momentum Equation

Let the speed of the SUV be denoted as \( v \). The initial momentum of the car is \( 1100 \times 32 \) and the initial momentum of the SUV is \( 2500 \times v \). Since they move in opposite directions, their momenta will subtract: \[ 1100 \times 32 = 2500 \times v \]
04

Solve for the SUV's Speed

Rearrange the equation to solve for \( v \):\[ v = \frac{1100 \times 32}{2500} \]
05

Calculate the Value

Calculate the expression:\[ v = \frac{1100 \times 32}{2500} = 14.08 \]Thus, the SUV was traveling at 14.08 m/s.

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Key Concepts

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

Head-on collision
A head-on collision is a type of crash where two vehicles hit each other front-to-front. This kind of collision often results in significant impact because both vehicles carry their entire momentum towards each other.
For this particular scenario, the colliding car and sport utility vehicle (SUV) come to a complete stop after the collision. This indicates that all the momentum they carried before the crash is redistributed, meaning their combined momenta cancel each other out.
In head-on collisions, understanding vehicle masses and velocities is crucial, as these directly influence the momentum and the resulting calculations.
Initial momentum
Initial momentum refers to the total momentum a system possesses before any event or interaction, such as a collision. In physics, momentum is calculated as the product of mass and velocity.
For the exercise at hand, the car's initial momentum is computed by multiplying its mass (1100 kg) by its velocity (32 m/s), resulting in a total of 35200 kg路m/s.
The sport utility vehicle also has an initial momentum defined by its mass and its velocity before the crash, denoted as \( v \). Since the vehicles are moving towards each other, their momenta will subtract, as one is positive and the other is negative due to their different directions.
Final total momentum
The final total momentum after the collision is zero because both vehicles come to a complete stop. According to the conservation of momentum, if no external forces interfere, the total momentum before the collision equals total momentum afterward.
This means that, in this scenario, the car's and SUV's initial momenta have fully cancelled each other out by the end of the collision. The equation illustrating this conservation is set up using all initial momenta, leading to the calculation of each vehicle's speed prior to stopping.
Opposite directions
In this exercise, the car and the SUV were moving in opposite directions before the collision. It is important to note the distinction because it determines how their momenta are calculated.
Opposite directions introduce the concept of positive and negative momentum. If one vehicle's momentum is considered positive, the opposite-moving vehicle鈥檚 momentum should be negative.
  • Car's momentum: positive \(1100 \times 32\)
  • SUV's momentum: negative \(2500 \times v\)

The subtraction of these momenta reflects their opposing directions and is essential for solving the exercise correctly. Such distinction ensures the principle of conservation of momentum can be applied accurately in calculations, leading to finding the SUV's initial speed.

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Most popular questions from this chapter

ssm Starting with an initial speed of \(5.00 \mathrm{~m} / \mathrm{s}\) at a height of \(0.300 \mathrm{~m}\), a \(1.50-\mathrm{kg}\) ball swings downward and strikes a \(4.60\) -kg ball that is at rest, as the drawing shows. (a) Using the principle of conservation of mechanical energy, find the speed of the \(1.50-\mathrm{kg}\) ball just before impact. (b) Assuming that the collision is elastic, find the velocities (magnitude and direction) of both balls just after the collision. (c) How high does each ball swing after the collision, ignoring air resistance?

A two-stage rocket moves in space at a constant velocity of \(4900 \mathrm{~m} / \mathrm{s} .\) The two stages are then separated by a small explosive charge placed between them. Immediately after the explosion the velocity of the 1200 -kg upper stage is \(5700 \mathrm{~m} / \mathrm{s}\) in the same direction as before the explosion. What is the velocity (magnitude and direction) of the 2400 -kg lower stage after the explosion?

Each of these problems consists of Concept Questions followed by a related quantitative Problem. The Concept Questions involve little or no mathematics. They focus on the concepts with which the problems deal. Recognizing the concepts is the essential initial step in any problem-solving technique. Concept Questions One object is at rest, and another is moving. Furthermore, one object is more massive than the other. The two collide in a one- dimensional completely inelastic collision. In other words, they stick together after the collision and move off with a common velocity. Momentum is conserved. (a) Consider the final momentum of the two-object system after the collision. Is it greater when the large-mass object is moving initially or when the small-mass object is moving initially? (b) Is the final speed after the collision greater when the large-mass or the smallmass object is moving initially? Problem The speed of the object that is moving initially is \(25 \mathrm{~m} / \mathrm{s}\). The masses of the two objects are 3.0 and \(8.0 \mathrm{~kg}\). Determine the final speed of the two-object system after the collision for the case when the large-mass object is moving initially and the case when the small-mass object is moving initially. Be sure that your answers are consistent with your answers to the Concept Questions.

ssm One average force \(\overrightarrow{\mathrm{F}}_{1}\) has a magnitude that is three times as large as that of another average force \(\overline{\mathrm{F}}_{2} .\) Both forces produce the same impulse. The average force \(\overrightarrow{\mathrm{F}}_{1}\) acts for a time interval of \(3.2 \mathrm{~ms}\). For what time interval does the average force \(\overrightarrow{\mathrm{F}}_{2}\) act?

Two people are standing on a 2.0 -m-long platform, one at each end. The platform floats parallel to the ground on a cushion of air, like a hovercraft. One person throws a \(6.0-\mathrm{kg}\) ball to the other, who catches it. The ball travels nearly horizontally. Excluding the ball, the total mass of the platform and people is \(118 \mathrm{~kg}\). Because of the throw, this 118 -kg mass recoils. How far does it move before coming to rest again?
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