Problem 5 - Applied Physics

Question

A \(98.0\) -kg parts cart with rubber bumpers rolling \(1.20 \mathrm{~m} / \mathrm{s}\) to the right crashes into a similar cart of mass \(125 \mathrm{~kg}\) moving left at \(0.750 \mathrm{~m} / \mathrm{s}\). After the collision, the lighter cart is traveling \(0.986 \mathrm{~m} / \mathrm{s}\) to the left. What is the velocity of the heavier cart after the collision?

Step-by-Step Solution

Answer

The heavier cart's velocity after the collision is approximately \(0.164\, \text{m/s}\) to the right.

Step 1: Understanding Conservation of Momentum

In this collision problem, the law of conservation of momentum applies because no external forces act on the system. The total momentum before the collision equals the total momentum after the collision.

Step 2: Write the Equation for Conservation of Momentum

Let the velocity of the lighter cart be denoted as \(v_1\) and of the heavier cart as \(v_2\) before the collision. Similarly, let \(v_1'\) and \(v_2'\) be their velocities after the collision. The equation then is: \[ m_1 v_1 + m_2 v_2 = m_1 v_1' + m_2 v_2' \] where \(m_1 = 98.0\, \text{kg}, m_2 = 125\, \text{kg}, v_1 = 1.20\, \text{m/s}, v_2 = -0.750\, \text{m/s}, v_1' = -0.986\, \text{m/s}\).

Step 3: Substitute Known Values into the Equation

Substitute the known values into the momentum equation: \[ 98.0 \times 1.20 + 125 \times (-0.750) = 98.0 \times (-0.986) + 125 \times v_2' \] Simplify the equation: \[ 117.6 - 93.75 = -96.628 + 125v_2' \]

Step 4: Solve for \(v_2'\)

Rearrange and solve for the unknown velocity \(v_2'\): \[ 125v_2' = 117.6 - 93.75 + 96.628 \] \[ 125v_2' = 20.478 \] \[ v_2' = \frac{20.478}{125} \] \[ v_2' \approx 0.1638 \, \text{m/s} \]

Key Concepts

Collision Problems

Collision problems are fascinating topics in physics education. They allow us to explore how objects interact when they collide. Whenever two objects collide, whether it's bumpers in a parts cart or cars on the road, certain predictable patterns occur.

In a collision, we primarily focus on:

• The type of collision: elastic or inelastic.
• Initial and final velocities of the involved objects.
• External forces: If absent, we apply the law of conservation of momentum.

Understanding these patterns helps simplify the complexity of collision problems. By knowing these key aspects, you can predict how objects behave after a collision even before you see it happen. This is why solving such problems is crucial for mastering physics.

Momentum Equation

The momentum equation is the backbone of any collision analysis.Momentum, a product of an object's mass and velocity, shows how much motion an object has.

The formula for linear momentum is given by:

• Momentun, \( p \), is calculated as \( p = m \times v \)
• Where \( m \) is mass and \( v \) is velocity.

For a system of objects, like the parts carts in our example, the conservation of momentum principle states that the total momentum before the collision is equal to the total momentum after the collision:

• Before collision: \( m_1 v_1 + m_2 v_2 \)
• After collision: \( m_1 v_1' + m_2 v_2' \)

This equation allows us to find the unknown velocities after a collision, as long as we know the masses and initial velocities.

Physics Education

Physics education becomes more insightful when hands-on problems like collision problems are examined. Real-life applications of concepts like the conservation of momentum deepen students' understanding and appreciation of physics.

Such problems encourage:

• Analytical thinking: Breaking down the problem into parts, like identifying known and unknown quantities.
• Problem-solving skills: Applying equations and mathematical skills to solve problems.
• Conceptual understanding: Grasping the broader implications of physical laws.

Use these opportunities to visualize collisions in real life, such as in sports, car accidents, or even planetary movements. This practical approach not only reinforces theoretical concepts but also makes learning more engaging and meaningful.

Velocity Calculation

Calculating velocity after a collision is an interesting and rewarding part of physics. Velocity, being a vector, has both magnitude and direction, which is essential in determining the final state of the objects involved in a collision.

In the context of collision problems:

• Initial and final velocities must be accurately determined to apply the momentum equation.
• Directions are crucial: Make sure to note which direction is considered positive; in our example, right is positive, so left is negative.

The calculation involves:

1. Using the conservation of momentum equation to relate the known and unknown velocities.
2. Solving for the unknown velocity using algebraic manipulation and substituting given values.
3. Interpreting the results, understanding what they mean in the context of the specific scenario.

By understanding this process, students can predict the outcome of collisions and apply these concepts to other physical scenarios.