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Chapter 8: Problem 21

On a frictionless, horizontal air table, puck \(A\) (with mass 0.250 kg) is moving toward puck \(B\) (with mass 0.350 kg), which is initially at rest. After the collision, puck A has a velocity of 0.120 m/s to the left, and puck \(B\) has a velocity of 0.650 m/s to the right. (a) What was the speed of puck \(A\) before the collision? (b) Calculate the change in the total kinetic energy of the system that occurs during the collision.

Short Answer

Expert verified
(a) 0.790 m/s; (b) -0.0023 J

Step by step solution

01

Understanding Momentum Conservation

The law of conservation of momentum states that the total momentum before the collision is equal to the total momentum after the collision. The formula for momentum is given by the mass of the object multiplied by its velocity. Therefore, for pucks A and B,\[ m_A \cdot v_{A_{initial}} + m_B \cdot v_{B_{initial}} = m_A \cdot v_{A_{final}} + m_B \cdot v_{B_{final}} \]where:- \( m_A = 0.250 \, \text{kg} \) (mass of puck A)- \( m_B = 0.350 \, \text{kg} \) (mass of puck B)- \( v_{B_{initial}} = 0 \, \text{m/s} \) (since puck B is initially at rest).
02

Substituting Known Values

Given that puck A has a final velocity \( v_{A_{final}} = -0.120 \, \text{m/s} \) (to the left) and puck B has a final velocity \( v_{B_{final}} = 0.650 \, \text{m/s} \) (to the right), substitute these values into the momentum equation:\[ 0.250 \cdot v_{A_{initial}} = 0.250 \cdot (-0.120) + 0.350 \cdot 0.650 \]
03

Calculating Initial Speed of Puck A

Simplify and solve for \( v_{A_{initial}} \):\[ 0.250 \cdot v_{A_{initial}} = -0.030 + 0.2275 \]\[ 0.250 \cdot v_{A_{initial}} = 0.1975 \]\[ v_{A_{initial}} = \frac{0.1975}{0.250} = 0.790 \, \text{m/s} \]
04

Understanding Kinetic Energy Change

The change in kinetic energy is found by calculating the total kinetic energy before the collision and the total kinetic energy after the collision, then finding the difference:\[ \Delta KE = KE_{final} - KE_{initial} \]
05

Calculating Initial and Final Kinetic Energies

The initial kinetic energy is only due to puck A, since puck B is at rest:\[ KE_{initial} = \frac{1}{2} m_A v_{A_{initial}}^2 = \frac{1}{2} \cdot 0.250 \cdot (0.790)^2 \]The final kinetic energies are:\[ KE_{A_{final}} = \frac{1}{2} \cdot 0.250 \cdot (-0.120)^2 \]\[ KE_{B_{final}} = \frac{1}{2} \cdot 0.350 \cdot (0.650)^2 \]
06

Calculating the Change in Kinetic Energy

Substitute and calculate the values:\[ KE_{initial} = 0.0780125 \text{ J} \]\[ KE_{A_{final}} = 0.0018 \text{ J} \]\[ KE_{B_{final}} = 0.0739375 \text{ J} \]Then,\[ KE_{final} = KE_{A_{final}} + KE_{B_{final}} = 0.0018 + 0.0739375 = 0.0757375 \text{ J} \]Finally, the change in kinetic energy:\[ \Delta KE = 0.0757375 - 0.0780125 = -0.002275 \text{ J} \]

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Key Concepts

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

Kinetic Energy
Kinetic energy is a type of energy that an object possesses due to its motion. You can calculate it using the formula \( KE = \frac{1}{2} m v^2 \), where \( m \) is the mass of the object and \( v \) is its velocity.
  • When an object is at rest, its kinetic energy is zero.
  • The faster an object moves, the greater its kinetic energy.
In the given exercise, before the collision, only puck A has kinetic energy. This is because puck B is initially at rest.
After the collision, both pucks have kinetic energy since they both have velocities. To find out how much energy is present before and after the collision, follow these steps:
  • Use the given velocities to calculate the kinetic energy for each puck.
  • Compare the total kinetic energy before and after to see how energy has changed during the collision.
Understanding kinetic energy helps to comprehend how energy moves from one object to another during interactions like collisions.
Elastic Collisions
Elastic collisions are very interesting because they obey certain specific laws of physics. In an elastic collision, two key properties are conserved:
  • Momentum
  • Kinetic energy
If a collision is elastic, the total kinetic energy of the system remains the same before and after the collision.
This isn't always the case in real-world scenarios. Often some kinetic energy is transformed into other forms of energy, like sound or heat, making most real-world collisions inelastic. For collisions that closely fit the ideal elastic model, like those between air-hockey pucks or billiard balls, these principles help to predict the behavior of the objects involved. In the given exercise, by calculating the initial and final kinetic energies, you found a slight decrease in kinetic energy, indicating it's not perfectly elastic. Nonetheless, understanding these principles aids in analyzing how objects interact when they collide.
Momentum Calculations
Momentum is a vector quantity, measured as the product of an object's mass and its velocity, represented by the formula:\[ p = m v \]In momentum calculations, the direction is crucial, as it determines positive or negative values depending on how the objects are moving.
  • Momentum follows the principle of conservation, meaning the total momentum before an event like a collision is equal to the total momentum afterward, assuming no external forces act on the system.
  • This principle allows us to solve for unknown values, such as the initial speed of an object.
In the context of the exercise, you applied the conservation of momentum to determine the initial speed of puck A before the collision by setting up an equation that equates the total initial and final momentum of both pucks. Understanding these calculations rules enables you to predict the outcomes of collisions in various physics problems.

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Most popular questions from this chapter

Experimental tests have shown that bone will rupture if it is subjected to a force density of 1.03 \(\times\) 10\(^8\) N/m\(^2\). Suppose a 70.0-kg person carelessly rollerskates into an overhead metal beam that hits his forehead and completely stops his forward motion. If the area of contact with the person's forehead is 1.5 cm\(^2\), what is the greatest speed with which he can hit the wall without breaking any bone if his head is in contact with the beam for 10.0 ms?

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The expanding gases that leave the muzzle of a rifle also contribute to the recoil. A .30-caliber bullet has mass 0.00720 kg and a speed of 601 m/s relative to the muzzle when fired from a rifle that has mass 2.80 kg. The loosely held rifle recoils at a speed of 1.85 m/s relative to the earth. Find the momentum of the propellant gases in a coordinate system attached to the earth as they leave the muzzle of the rifle.
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