Problem 29 - College Physics

Question

A man of mass \(m_{1}=70.0 \mathrm{~kg}\) is skating at \(v_{1}=\) \(8.00 \mathrm{~m} / \mathrm{s}\) behind his wife of mass \(\mathrm{m}_{2}=50.0 \mathrm{~kg}\), who is skating at \(\tau_{2}=4.00 \mathrm{~m} / \mathrm{s}\). Instead of passing her, he inadvertently collides with her. He grabs her around the waist, and they maintain their balance. (a) Sketch the problem with before-and-after diagrams, representing the skaters as blocks. (b) Is the collision best described as elastic, inelastic, or perfectly inelastic? Why? (c) Write the general equation for conservation of momentum in terms of \(m_{1}, v_{1}, w_{2}, v_{2}\), and final velocity \(v\) - (d) Solve the momentum equation for \(v_{\gamma}\). (e) Substitute values, obtaining the numerical value for \(v_{f}\), their speed after the collision.

Step-by-Step Solution

Answer

The collision is perfectly inelastic. The final velocity after the collision, found by applying the principle of conservation of momentum, is \(v_f = 6.00 m/s\).

Step 1: Analyze the Collision Type

In this case, since the man grabs his wife, they move together post the collision which means their speeds are the same. This identifies the collision as perfectly inelastic.

Step 2: Understand Conservation of Momentum Principle

The principle of conservation of momentum states that the total momentum before the collision is equal to the total momentum after the collision. We represent it mathematically by \(m_{1}v_{1} + m_{2}v_{2} = (m_{1}+m_{2})v_f\), where \(v_f\) is their common final speed.

Step 3: Apply Conservation of Momentum Principle to Find Final Velocity

To find the final velocity \(v_f\) after the collision, we rearrange the formula: \(v_f = (m_{1}v_{1} + m_{2}v_{2}) /(m_{1}+m_{2})\)

Step 4: Substitute the values and find the Result

Substituting the values, we get \(v_f = (70.0 kg * 8.00 m/s + 50.0 kg * 4.00 m/s) / (70.0 kg + 50.0 kg)\). Solving this equation gives the final velocity after the collision.

Key Concepts

Perfectly Inelastic Collision

In the realm of classical mechanics, a perfectly inelastic collision is a scenario where two colliding objects move together post-collision, implying they combine velocities and thus, share a single speed. Often, this involves the objects sticking together, as seen in our example where the man grabs his wife around the waist after colliding. In such events, kinetic energy is not conserved, while momentum is. This means while they lose some movement energy (turned into other forms like heat or deformation), their shared momentum before and after remains constant. It’s a quintessential example of how energy and momentum behave under different collision types.

Physics Problem Solving

When approaching physics problems, especially those involving collisions, it’s vital to methodically analyze the situation. We start by identifying the given parameters and the type of collision involved. First, sketching the problem scenario helps visualize dynamics clearly.
Next, look for conservation laws applicable to the situation, such as conservation of momentum in collision problems. Simplify complex scenarios by breaking them into stepwise processes:

• Recognize the collision type.
• Apply the proper equations.
• Solve the equations meticulously.
• Substitute the known values to find the unknowns.

By following these steps, you build a solid foundation for approaching both basic and complex problems efficiently.

Momentum Equation

The principle of conservation of momentum is foundational in physics. It is expressed in the momentum equation, \[m_{1}v_{1} + m_{2}v_{2} = (m_{1}+m_{2})v_f\]. This equation states that the sum of initial momenta, represented as the product of mass and velocity for the individual bodies, equals the final momentum when they act as a single entity post-collision.
To find the final velocity \(v_f\), rearrange the equation:\[v_f = \frac{m_{1}v_{1} + m_{2}v_{2}}{m_{1}+m_{2}}\] Here, each object's contribution to the total momentum is summed up to determine the shared speed after impact. It’s crucial to remember, this principle applies broadly across various collision types in an isolated system.

Collision Types

Understanding collision types is critical in physics to determine how objects interact post-impact. They are generally categorized into three main types:

• Elastic Collisions: Both momentum and kinetic energy are conserved. Objects rebound without deformation.
• Inelastic Collisions: Momentum is conserved but kinetic energy is not. The objects might touch but not stick together.
• Perfectly Inelastic Collisions: A subset of inelastic collisions where the objects stick together post-collision, thus moving with the same velocity.

Each type outlines different outcomes for speed and kinetic energy after a collision. Recognizing these differences helps in analyzing collision problems accurately, paving the way for correctly applying the corresponding physics formulas.