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Chapter 7: Problem 14

It is well known that bullets and other missiles fired at Superman simply bounce off his chest (Fig.7-22). Suppose that a gangster sprays Superman's chest with \(3 \mathrm{~g}\) bullets at the rate of 100 bullets/min, and the speed of each bullet is \(500 \mathrm{~m} / \mathrm{s}\). Suppose too that the bullets rebound straight back with no change in speed. What is the magnitude of the average force on Superman's chest from the stream of bullets?

Short Answer

Expert verified
The magnitude of the average force on Superman's chest is 5 N.

Step by step solution

01

Understand the Problem

Determine the initial and final momentum, the rate of change of momentum, and how to use these to find the force applied to Superman's chest.
02

Calculate the Momentum of One Bullet

Use the formula for momentum: \[ p = mv \] where \(m = 3 \text{ g} = 0.003 \text{ kg}\) and \(v = 500 \text{ m/s}\). Therefore, \[ p = 0.003 \times 500 = 1.5 \text{ kg鈰卪/s} \]
03

Determine the Change in Momentum for One Bullet

Since the bullet rebounds with no change in speed, it has an initial momentum of \(1.5 \text{ kg鈰卪/s}\) and a final momentum of \(-1.5 \text{ kg鈰卪/s}\). The change in momentum \( \triangle p \) is calculated as: \[ \triangle p = p_{final} - p_{initial} = -1.5 - 1.5 = -3 \text{ kg鈰卪/s} \]
04

Calculate the Total Change in Momentum per Minute

Given the rate of 100 bullets per minute, the total change in momentum per minute is: \[ \triangle p_{total} = 100 \times (-3) = -300 \text{ kg鈰卪/s} \]
05

Convert Time to Seconds and Calculate Average Force

1 minute is 60 seconds. The average force, \( F_{avg} \), can be calculated using the formula: \[ F_{avg} = \frac{ \triangle p_{total} }{ \triangle t } = \frac{ -300 }{ 60 } = -5 \text{ N} \] The magnitude of the force is \( 5 \text{ N} \).

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Key Concepts

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

momentum
Momentum is a fundamental concept in physics, describing the quantity of motion an object has. Momentum is defined as the product of an object's mass and velocity, represented by the formula: \( p = mv \). Here, 'p' is momentum, 'm' is mass, and 'v' is velocity.

In the problem, each bullet has a mass of \(3 \text{ g} \) or \(0.003 \text{ kg} \) and a speed of \(500 \text{ m/s} \). Using the momentum formula, we calculated the momentum of one bullet to be: \[ p = 0.003 \times 500 = 1.5 \text{ kg鈰卪/s} \]

Understanding momentum helps us analyze changes in motion when objects interact. In this scenario, the gangster鈥檚 bullets hit and bounce back from Superman's chest without slowing down, leading to a reversal in momentum.
force calculation
Force is what causes an object to accelerate, and it is calculated using Newton鈥檚 second law of motion, which states that force is equal to the rate of change of momentum. The formula is: \( F = \frac{ \triangle p }{ \triangle t } \) where \( \triangle p \) is the change in momentum and \( \triangle t \) is the time interval.

In the exercise, each bullet changes its momentum when it bounces back. The initial momentum is \(1.5 \text{ kg鈰卪/s} \) and the final momentum is \(-1.5 \text{ kg鈰卪/s} \). Therefore, the change in momentum for one bullet is: \[ \triangle p = p_{ \text{ final }} \text{ - } p_{ \text{ initial }} \text{ = } (-1.5) - 1.5 = -3 \text{ kg鈰卪/s} \]

Given 100 bullets per minute, the total change in momentum per minute is \(-300 \text{ kg鈰卪/s} \). Converting this into seconds, the average force on Superman鈥檚 chest is: \[ F_{ \text{ avg }} = \frac{ -300 }{ 60 } = -5 \text{ N} \] The magnitude of which is \(5 \text{ N} \).
projectile motion
Projectile motion describes the motion of an object thrown or projected into the air, subject to only the acceleration of gravity. In this exercise, bullets are moving towards Superman's chest, and they follow a linear path due to the high speed and close range.

For projectile motion, key factors include initial velocity, angle of projection, and effects of gravity. Even if the bullets aren't falling freely, understanding projectile motion helps grasp how objects move when launched. Each bullet has a speed of \(500 \text{ m/s} \) and travels in a straight line towards Superman before rebounding.

Factors like air resistance and trajectory can influence projectile motion, but with bullets and a direct path, these effects are minimal. Hence, this problem simplifies to linear motion with a rebound effect, aligning with fundamental projectile motion concepts.

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Most popular questions from this chapter

In a game of pool, the cue ball strikes another ball of the same mass and initially at rest. After the collision, the cue ball moves at \(3.50 \mathrm{~m} / \mathrm{s}\) along a line making an angle of \(22.0^{\circ}\) with its original direction of motion, and the second ball has a speed of \(2.00 \mathrm{~m} / \mathrm{s}\). Find (a) the angle between the direction of motion of the second ball and the original direction of motion of the cue ball and (b) the original speed of the cue ball.

In February 1955, a paratrooper fell \(370 \mathrm{~m}\) from an airplane without being able to open his chute but happened to land in snow, suffering only minor injuries. Assume that his speed at impact was \(56 \mathrm{~m} / \mathrm{s}\) (terminal speed), that his mass (including gear) was \(85 \mathrm{~kg}\), and that the magnitude of the force on him from the snow was at the survivable limit of \(1.2 \times 10^{5} \mathrm{~N}\). What are (a) the minimum depth of snow that would have stopped him safely and (b) the magnitude of the impulse on him from the snow?

When jumping straight down, you can be seriously injured if you land stiff- legged. One way to avoid injury is to bend your knees upon landing to reduce the force of the impact. Suppose you have a mass \(m\) and you jump off a wall of height \(h\). (a) Use what you learned about constant acceleration motion to find the speed with which you hit the ground. Assume you simply step off the wall, so your initial \(y\) velocity is zero. Ignore air resistance. (Express your answer in terms of the symbols given.) (b) Suppose that the time interval starting when your feet first touch the ground until you stop is \(\Delta t .\) Calculate the (average) net force acting on you during that interval. (Again, express your answer in terms of the symbols given.) (c) Suppose \(h=1 \mathrm{~m}\). If you land stiff-legged, the time it takes you to stop may be as short as \(2 \mathrm{~ms}\), whereas if you bend your knees, it might be as long as \(0.1\) s. Calculate the average net force that would act on you in the two cases. (d) The net force on you while you are stopping includes both the force of gravity and the force of the ground pushing up. Which of these forces do you think does you the injury? Explain your reasoning. (e) For the two cases in part (c), calculate the upward force the ground exerts on you.

An alpha particle collides with an oxygen nucleus that is initially at rest. The alpha particle is scattered at an angle of \(64.0^{\circ}\) from its initial direction of motion, and the oxygen nucleus recoils at an angle of \(51.0^{\circ}\) on the opposite side of that initial direction. The final speed of the nucleus is \(1.20 \times 10^{5} \mathrm{~m} / \mathrm{s}\). Find (a) the final speed and (b) the initial speed of the alpha particle. (In atomic mass units, the mass of an alpha particle is \(4.0 \mathrm{u}\), and the mass of an oxygen nucleus is \(16 \mathrm{u}\).)

A professor of physics is going ice skating for the first time. He has gotten himself into the middle of an ice rink and cannot figure out how to make the skates work. Every motion he makes simply causes his feet to slip on the ice and leaves him in the same place he started. He decides that he can get off the ice by throwing his gloves in the opposite direction. (a) Suppose he has a mass \(M\) and his gloves have a mass \(m\). If he throws the gloves as hard as he can away from him, they leave his hand with a velocity \(\vec{v}_{\text {glove }}\) Explain whether or not he will move. If he does move, calculate his velocity, \(\vec{v}_{\text {prof }}\) (b) Discuss his motion from the point of view of the forces acting on him. (c) If the ice rink is \(10 \mathrm{~m}\) in diameter and the skater starts in the center, estimate how long it will take him to reach the edge, assuming there is no friction at all.
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