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Chapter 9: Problem 72

\(\|\) At the county fair, Chris throws a \(0.15 \mathrm{kg}\) baseball at a \(2.0 \mathrm{kg}\) wooden milk bottle, hoping to knock it off its stand and win a prize. The ball bounces straight back at \(20 \%\) of its incoming speed, knocking the bottle straight forward. What is the bottle's speed, as a percentage of the ball's incoming speed?

Short Answer

Expert verified
The final speed of the wooden milk bottle is 3% of the incoming speed of the baseball.

Step by step solution

01

Understand the problem

The ball has an initial mass \(m_1 = 0.15kg\) and an unknown initial speed \(v_{1i}\). After hitting the bottle, it bounces back with \(20\%\) of its initial speed, so the final speed \(v_{1f} = -0.2v_{1i}\) is negative because it鈥檚 going in the opposite direction. The wooden milk bottle has a mass of \(m_2 = 2kg\), its initial speed \(v_{2i} = 0\) (since it's stationary) and an unknown final velocity \(v_{2f}\). The question asks for the value of \(v_{2f}\) as a percentage of the initial speed of the ball \(v_{1i}\).
02

Using the principle of conservation of momentum

The principle of conservation of momentum states that the total momentum of a system of objects is constant if no external forces are acting upon it. In this case, we have no external forces acting on the ball and the bottle. So, we have:\[m_{1}v_{1i} + m_{2}v_{2i} = m_{1}v_{1f} + m_{2}v_{2f}\]Substituting the known values we have: \(0.15v_{1i} = 0.15(-0.2v_{1i}) + 2v_{2f}\). Solving the equation for \(v_{2f}\) (final speed of the bottle), we get: \(v_{2f} = 0.03v_{1i}\).
03

Find the final speed as a percentage

To find the final speed of the bottle as a percentage of the ball鈥檚 initial speed, just multiply the ratio by 100:\[Percentage = 100 \times v_{2f} / v_{1i} = 100 \times 0.03 = 3\%\]This means that the final speed of the bottle is 3% of the ball's initial speed.

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Key Concepts

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

Momentum
The concept of momentum in physics is fundamental to understanding various phenomena, especially when it comes to motion and collisions. Momentum, symbolized by the letter 'p', is the product of an object's mass (m) and its velocity (v). Hence, the momentum (\( p \) is given by the equation \( p = m \times v \)em>. In the context of our exercise, both the baseball and the wooden milk bottle possess momentum. The key feature of momentum is that in an isolated system (where no external forces act on the system), the total momentum before and after an event, like a collision, remains conserved.

When Chris throws the baseball at the bottle, the system consists of the ball and bottle. Initially, the bottle is at rest, carrying no momentum, while the moving ball has a certain amount of momentum. After collision, the ball reverses its direction, and the bottle starts moving, indicating a transfer of momentum has occurred. Applying this principle of conservation of momentum allows us to predict the motion of objects post collision, as seen in the step-by-step solution provided.
Inelastic Collision
In an inelastic collision, two objects collide and do not maintain their individual shapes or energy; instead, they lose some kinetic energy during the process. This kind of collision contrasts with a perfectly elastic collision, where there's no net loss in kinetic energy within the system. Although the ball in the problem doesn't stick to the bottle, which would make it a perfectly inelastic collision, it is still considered inelastic because the ball loses speed after hitting the bottle.

Despite the energy loss, the principle of conservation of momentum still holds true. This exercise exemplifies inelastic collision by showing that the total momentum before and after the collision remains the same but does not consider the kinetic energy transferred to sound, heat, and deformation. The question challenges students to determine the change in momentum by comparing velocities, providing practice in associating inelastic collisions with momentum conservation.
Physics Problem-Solving
Physics problems often require a structured approach to arrive at the correct solution. The problem featuring Chris and the baseball employs a sequential method to problem-solving. This method involves:
  • Understand the Problem: Identifying known variables and what needs to be solved.
  • Implement Physics Principles: Applying laws, such as the conservation of momentum.
  • Mathematical Solution: Manipulating equations to reach a numerical solution.

In this scenario, understanding the problem entails recognizing that the 'collision' event is key and that the 'speed' of each object will change as a result. The use of conservation of momentum is critical to formulating an equation with the known and unknown variables. Finally, algebraic manipulation leads to finding the unknown speed as a percentage of the ball's initial speed, which cultivates the students鈥� ability to transition from conceptual understanding to practical application.

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Most popular questions from this chapter

Peregrine falcons frequently grab prey birds from the air, as in Example 9.10. Sometimes they strike at high enough speeds that the force of the impact disables prey birds. A 480 g peregrine falcon high in the sky spies a 240 g pigeon some distance below. The falcon slows to a near stop, then goes into a dive \(-\) called a stoop - and picks up speed as she falls. The falcon reaches a vertical speed of \(45 \mathrm{m} / \mathrm{s}\) before striking the pigeon, which we can assume is stationary. The falcon strikes the pigeon and grabs it in her talons. The collision between the birds lasts 0.015 s. a. What is the final speed of the falcon and pigeon? b. What is the average force on the pigeon during the impact?

Figure \(\mathrm{P} 9.75\) shows a \(100 \mathrm{g}\) puck revolving at \(100 \mathrm{rpm}\) on a 20-cm-radius circle on a frictionless table. A string attached to the puck passes through a hole in the middle of the table. The end of the string below the table is then slowly pulled down until the puck is revolving in a 10 -cm-radius circle. How many revolutions per minute does the puck make at this new radius?

In a Little League baseball game, the 145 g ball reaches the batter with a speed of \(15.0 \mathrm{m} / \mathrm{s}\). The batter hits the ball, and it leaves his bat with a speed of \(20.0 \mathrm{m} / \mathrm{s}\) in exactly the opposite direction. What is the magnitude of the impulse delivered by the bat to the ball? b. If the bat is in contact with the ball for \(1.5 \mathrm{ms},\) what is the magnitude of the average force exerted by the bat on the ball?

A 15 g bullet is fired at \(610 \mathrm{m} / \mathrm{s}\) into a \(4.0 \mathrm{kg}\) block that sits at the edge of a \(75-\mathrm{cm}\) -high table. The bullet embeds itself in the block and carries it off the table. How far from the point directly below the table's edge does the block land?

Ferns spread spores instead of seeds, and some ferns eject the spores at surprisingly high speeds. One species accelerates \(1.4 \mu \mathrm{g}\) spores to a \(4.5 \mathrm{m} / \mathrm{s}\) ejection speed in a time of \(1.0 \mathrm{ms}\) What impulse is provided to the spores? What is the average force on a spore?
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