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Chapter 7: Problem 4

In a performance test, each of two cars takes 9.0 s to accelerate from rest to 27 m/s. Car A has a mass of 1400 kg, and car B has a mass of 1900 kg. Find the net average force that acts on each car during the test.

Short Answer

Expert verified
The net average force on Car A is 4200 N; on Car B, it is 5700 N.

Step by step solution

01

Identify Given Information

We are given that both cars take 9.0 seconds to accelerate from rest to 27 m/s. Car A has a mass of 1400 kg, and Car B has a mass of 1900 kg.
02

Calculate Acceleration

To find acceleration, we use the formula \( a = \frac{\Delta v}{\Delta t} \), where \( \Delta v = 27 \text{ m/s} \) and \( \Delta t = 9.0 \text{ s} \). So, \( a = \frac{27 \text{ m/s}}{9.0 \text{ s}} = 3 \text{ m/s}^2 \).
03

Apply Newton's Second Law for Car A

To find the net average force on Car A, use \( F = ma \). For Car A, \( m = 1400 \text{ kg} \) and \( a = 3 \text{ m/s}^2 \). Therefore, \( F_A = 1400 \text{ kg} \times 3 \text{ m/s}^2 = 4200 \text{ N} \).
04

Apply Newton鈥檚 Second Law for Car B

Similarly, for Car B, use the same formula \( F = ma \). For Car B, \( m = 1900 \text{ kg} \). Therefore, \( F_B = 1900 \text{ kg} \times 3 \text{ m/s}^2 = 5700 \text{ N} \).
05

Conclusion

The net average force acting on Car A is 4200 N, and the net average force acting on Car B is 5700 N.

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Key Concepts

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

Acceleration in Motion
Acceleration is a fundamental concept in physics that describes how an object's velocity changes over time. It's calculated as the change in velocity divided by the time over which the change occurs. In this scenario, two cars accelerate from 0 to 27 m/s in 9.0 seconds. To determine the acceleration, we use the formula:
  • \( a = \frac{\Delta v}{\Delta t} \)
  • where \( \Delta v \) is the change in velocity and \( \Delta t \) is the change in time.
Plugging in the values, \( a = \frac{27 \text{ m/s}}{9.0 \text{ s}} = 3 \text{ m/s}^2 \). This means both cars have a constant acceleration of 3 m/s虏 throughout the test. Acceleration helps us understand how quickly the cars are gaining speed. This rate is a critical part of analyzing any kind of motion.
Understanding Net Force
The net force is the total force acting on an object when all the individual forces are combined. According to Newton's Second Law, the net force is directly related to the mass and acceleration of an object. It is expressed as:
  • \( F = ma \)
  • where \( F \) is the net force, \( m \) is the mass, and \( a \) is the acceleration.
For Car A, with a mass of 1400 kg and acceleration of 3 m/s虏, the net force is:
  • \( F_A = 1400 \times 3 = 4200 \text{ N} \).
Similarly, for Car B, with a mass of 1900 kg, the net force is:
  • \( F_B = 1900 \times 3 = 5700 \text{ N} \).
In essence, net force gives us the push or pull needed to maintain the acceleration of an object with a given mass. It is pivotal to understanding how forces affect motion.
The Role of Mass
Mass is a measure of the amount of matter in an object, and it plays a critical role in motion. In the context of this exercise, the mass of each car influences how much force is required to accelerate them. Newton's Second Law highlights this relationship with the formula:
  • \( F = ma \)
The greater the mass of an object, the more force is needed to achieve the same acceleration. For example, Car B has a higher mass (1900 kg) compared to Car A (1400 kg), thus requiring a greater net force (5700 N for Car B vs. 4200 N for Car A) to reach the same acceleration. Understanding mass helps in analyzing how different objects behave under similar forces.
Kinematics and Motion Analysis
Kinematics is the branch of mechanics that describes the motion of objects without considering the forces that cause the motion. It involves the analysis of various motion aspects, such as velocity, acceleration, and position over time. In this exercise, the kinematic analysis helps us understand the cars' journey from rest to a speed of 27 m/s. Kinematics provides a framework for analyzing scenarios like this:
  • It tells us how objects move, providing details about velocity and acceleration.
  • By using kinematic equations, we can predict future positions and velocities.
Overall, kinematics is essential for understanding and predicting the motions of everyday objects without delving into force interactions directly.

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Most popular questions from this chapter

In a science fiction novel two enemies, Bonzo and Ender, are fighting in outer space. From stationary positions they push against each other. Bonzo flies off with a velocity of + 11.5 m/s, while Ender recoils with a velocity of \(-2.5 \mathrm{m} / \mathrm{s}\). (a) Without doing any calculations, decide which person has the greater mass. Give your reasoning. (b) Determine the ratio \(m_{\text {Boano }} / m_{\text {Ender }}\) of the masses of these two enemies.

John's mass is \(86 \mathrm{kg}\), and Barbara's is \(55 \mathrm{kg}\). He is standing on the \(x\) axis at \(x_{1}=+9.0 \mathrm{m},\) while she is standing on the \(x\) axis at \(x_{\mathrm{B}}=+2.0 \mathrm{m}\) They switch positions. How far and in which direction does their center of mass move as a result of the switch?

A girl is skipping stones across a lake. One of the stones accidentally ricochets off a toy boat that is initially at rest in the water (see the drawing). The \(0.072-\mathrm{kg}\) stone strikes the boat at a velocity of \(13 \mathrm{m} / \mathrm{s}, 15^{\circ}\) below due east, and ricochets off at a velocity of \(11 \mathrm{m} / \mathrm{s}, 12^{\circ}\) above due east. After being struck by the stone, the boat's velocity is \(2.1 \mathrm{m} / \mathrm{s}\), due east. What is the mass of the boat? Assume the water offers no resistance to the boat's motion.

A 4.00-g bullet is moving horizontally with a velocity of \(+355 \mathrm{m} / \mathrm{s},\) where the \(+\) sign indicates that it is moving to the right (see part \(a\) of the drawing). The bullet is approaching two blocks resting on a horizontal frictionless surface. Air resistance is negligible. The bullet passes completely through the first block (an inelastic collision) and embeds itself in the second one, as indicated in part \(b\). Note that both blocks are moving after the collision with the bullet. The mass of the first block is \(1150 \mathrm{g}\), and its velocity is \(+0.550 \mathrm{m} / \mathrm{s}\) after the bullet passes through it. The mass of the second block is \(1530 \mathrm{g}\). (a) What is the velocity of the second block after the bullet embeds itself? (b) Find the ratio of the total kinetic energy after the collisions to that before the collisions.

Two arrows are fi red horizontally with the same speed of 30.0 m/s. Each arrow has a mass of 0.100 kg. One is fi red due east and the other due south. Find the magnitude and direction of the total momentum of this two-arrow system. Specify the direction with respect to due east.
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