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Chapter 7: Problem 45

Consult Concept Simulation \(7.1\) at in preparation for this problem. Two friends, \(\mathrm{Al}\) and Jo, have a combined mass of \(168 \mathrm{~kg}\). At an ice skating rink they stand close together on skates, at rest and facing each other, with a compressed spring between them. The spring is kept from pushing them apart because they are holding each other. When they release their arms, Al moves off in one direction at a speed of \(0.90 \mathrm{~m} / \mathrm{s}\), while Jo moves off in the opposite direction at a speed of \(1.2 \mathrm{~m} / \mathrm{s}\). Assuming that friction is negligible, find Al's mass.

Short Answer

Expert verified
Al's mass is 96 kg.

Step by step solution

01

Understand the Problem

This problem involves the conservation of linear momentum. When Al and Jo release the spring, they move apart due to the reactionary force exerted by the spring. The total momentum before they separate is zero because they are initially at rest, and we must use this information to find Al's mass.
02

Apply Conservation of Momentum

According to the conservation of momentum, the total momentum before and after they release the spring must be the same. Initially, the momentum is zero because they are at rest. After separation, the total momentum is the sum of the momenta of Al and Jo. Therefore, we have the equation: \( m_{Al} \cdot v_{Al} + m_{Jo} \cdot v_{Jo} = 0 \), where \( m_{Al} \) and \( v_{Al} \) are Al's mass and velocity, and \( m_{Jo} \) and \( v_{Jo} \) are Jo's mass and velocity. Note that Jo's velocity should be taken as negative because she moves in the opposite direction of Al.
03

Express Jo's Mass in terms of Al's Mass

The combined mass of Al and Jo is given as 168 kg. We can express Jo's mass in terms of Al's mass: \( m_{Jo} = 168 - m_{Al} \). This allows us to substitute \( m_{Jo} \) in the momentum equation.
04

Set Up the Equations

Substitute Jo's mass into the conservation equation: \( m_{Al} \cdot 0.9 + (168 - m_{Al}) \cdot (-1.2) = 0 \). Simplify the expression to isolate \( m_{Al} \): \( 0.9m_{Al} - 1.2(168) + 1.2m_{Al} = 0 \).
05

Solve for Al's Mass

Combine like terms to get \( 2.1m_{Al} = 1.2 \times 168 \). This simplifies to \( m_{Al} = \frac{1.2 \times 168}{2.1} \). Calculate \( m_{Al} \) to find that Al's mass is 96 kg.

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Key Concepts

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

Understanding Linear Momentum
Linear momentum is a fundamental concept in physics. It describes the motion of an object and is calculated as the product of its mass and velocity. Mathematically, linear momentum is expressed as:\[ p = m \cdot v \]where:
  • \( p \) is the momentum.
  • \( m \) is the mass of the object.
  • \( v \) is the velocity of the object.
Momentum is a vector quantity, meaning it has both magnitude and direction. In our ice skating problem, when Al and Jo push apart, they acquire momentum in opposite directions. This helps illustrate the concept that an object's momentum depends on both its mass and its velocity.
Approaching Physics Problem Solving
Solving physics problems, like the one involving Al and Jo, requires a methodical approach. First, it's important to understand what the problem is asking. Identify the known and unknown variables. In this problem, we know the combined mass of Al and Jo, their velocities after release, and we need to find Al鈥檚 mass.
Next, select the relevant concepts or laws that apply to the problem. Here, conservation of momentum is key because it tells us that the total momentum before and after their separation remains constant. Set up equations based on these principles.
Finally, simplify and solve these equations. Being systematic and organizing your work step by step is crucial for accuracy in physics problem solving.
Calculating Mass Accurately
Mass calculation in physics often involves rearranging equations to isolate the variable of interest. In this scenario, we're interested in finding Al's mass. Given that the combined mass of Al and Jo is 168 kg, we use this to reflect Jo鈥檚 mass regarding Al's:
  • \( m_{Jo} = 168 - m_{Al} \)
Incorporate this into the conservation of momentum equation and solve for Al鈥檚 mass. Accuracy in calculating mass is vital as incorrect computations can lead to wrong results down the line in your physics problems.
Understanding Velocity's Role
Velocity plays a crucial role in understanding momentum. It is a vector quantity, which means it has direction as well as magnitude. In Al and Jo's problem, when the spring is released:
  • Al moves with a velocity of \( 0.90 \text{ m/s} \).
  • Jo moves with a velocity of \( 1.20 \text{ m/s} \), but in the opposite direction.
The direction of these velocities is significant because, for momentum calculations, one direction is usually considered positive. The other is negative due to their opposite paths. Properly accounting for the signs when summing momentum ensures the accuracy of your calculations in physics problems.

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Most popular questions from this chapter

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 As the drawing illustrates, two disks with masses \(m_{1}\) and \(m_{2}\) are moving horizontally to the right at a speed \(v_{0}\). They are on an air-hockey table, which supports them with an essentially frictionless cushion of air. They move as a unit, with a compressed spring between them, which has a negligible mass. (a) When the spring is released and allowed to push outward, what are the directions of the forces that act on disk 1 and disk \(2 ?\) (b) After the spring is released, is the speed of each disk larger than, smaller than, or the same as the speed \(v_{0}\) ? Explain. Problem Consider the situation where disk 1 comes to a momentary halt shortly after the spring is released. Assuming that \(m_{1}=1.2 \mathrm{~kg}, m_{2}=2.4 \mathrm{~kg}\), and \(v_{0}=5.0 \mathrm{~m} / \mathrm{s},\) find the speed of disk 2 at that moment. Verify that your answer is consistent with your answers to the Concept Questions.

A ball is dropped from rest at the top of a \(6.10\) -m-tall building, falls straight downward. collides inelastically with the ground, and bounces back. The ball loses \(10.0 \%\) of its kinetic energy every time it collides with the ground. How many bounces can the ball make and still reach a windowsill that is \(2.44 \mathrm{~m}\) above the ground?

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At illustrates the physics principles in this problem. An astronaut in his space suit and with a propulsion unit (empty of its gas propellant) strapped to his back has a mass of \(146 \mathrm{~kg}\). During a space-walk, the unit, which has been completely filled with propellant gas, ejects some gas with a velocity of \(+32 \mathrm{~m} / \mathrm{s}\). As a result, the astronaut recoils with a velocity of \(-0.39 \mathrm{~m} / \mathrm{s}\). After the gas is ejected, the mass of the astronaut (now wearing a partially empty propulsion unit) is \(165 \mathrm{~kg}\). What percentage of the gas propellant in the completely filled propulsion unit was depleted?

During July 1994 the comet Shoemaker-Levy 9 smashed into Jupiter in a spectacular fashion. The comet actually consisted of 21 distinct pieces, the largest of which had a mass of approximately \(4.0 \times 10^{12} \mathrm{~kg}\) and a speed of \(6.0 \times 10^{4} \mathrm{~m} / \mathrm{s}\). Jupiter, the largest planet in the solar system, has a mass of \(1.9 \times 10^{27} \mathrm{~kg}\) and an orbital speed of \(1.3 \times 10^{4} \mathrm{~m} / \mathrm{s} .\) If this piece of the comet had hit Jupiter head-on, what would have been the change (magnitude only) in Jupiter's orbital speed (not its final speed)?
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