Problem 4 - Applied Physics

Question

A \(90.0\) -g disk traveling \(3.00 \mathrm{~m} / \mathrm{s}\) to the right collides with a \(75.0\) -g disk traveling \(8.00 \mathrm{~m} / \mathrm{s}\) to the left. After the collision, the heavier disk is traveling \(7.00 \mathrm{~m} / \mathrm{s}\) to the left. What is the velocity of the lighter disk after the collision?

Step-by-Step Solution

Answer

The lighter disk's velocity after the collision is 4.00 m/s to the right.

Step 1: Understand the Concept

This is a perfectly inelastic collision problem where we use the law of conservation of momentum. The total momentum before the collision is equal to the total momentum after the collision.

Step 2: Identify the Given Variables

We have the masses and initial velocities of two disks:- Mass of disk 1 (heavier): \(m_1 = 90.0\,\text{g} = 0.090\,\text{kg}\)- Initial velocity of disk 1: \(v_{1i} = 3.00\,\text{m/s}\)- Mass of disk 2 (lighter): \(m_2 = 75.0\,\text{g} = 0.075\,\text{kg}\)- Initial velocity of disk 2: \(v_{2i} = -8.00\,\text{m/s}\)\ (since it is traveling to the left)

Step 3: Define the Unknown Variable

We need to find the final velocity of disk 2 (lighter disk) after the collision: \(v_{2f}\). We are also given that the final velocity of the heavier disk is \(v_{1f} = -7.00\,\text{m/s}\).

Step 4: Apply Conservation of Momentum

The conservation of momentum formula for two objects in an isolated system is:\[m_1 v_{1i} + m_2 v_{2i} = m_1 v_{1f} + m_2 v_{2f}\]

Step 5: Substitute Known Values into the Equation

Substitute the known values into the momentum equation:\[(0.090)(3.00) + (0.075)(-8.00) = (0.090)(-7.00) + (0.075)v_{2f}\]

Step 6: Solve for Final Velocity of Lighter Disk

Calculate each term and solve for \(v_{2f}\):\[0.270 - 0.600 = -0.630 + 0.075v_{2f}\]Simplifying, \[-0.330 = -0.630 + 0.075v_{2f}\]add 0.630 to both sides:\[0.300 = 0.075v_{2f}\]Divide both sides by 0.075:\[v_{2f} = \frac{0.300}{0.075} = 4.00\,\text{m/s}\]

Step 7: Conclusion

The velocity of the lighter disk after the collision is \(4.00\,\text{m/s}\) to the right.

Key Concepts

Understanding Inelastic Collisions

In everyday terms, a collision refers to when two or more objects bump into each other. An inelastic collision is a specific type of collision where the colliding objects stick together or affect each other significantly, exchanging energy and momentum. Crucially, in a perfectly inelastic collision, kinetic energy is not conserved after the collision, but momentum is.
In the context of our problem, two disks collide with each other on a surface. The essential property of an inelastic collision here is that the velocity of just one disk changes, reflecting how they impacted each other and ensuring momentum conservation.

Applying the Momentum Equation

Momentum is a measurement of an object's motion and is calculated by multiplying an object's mass by its velocity. The principle of conservation of momentum states that the total momentum of an isolated system remains constant if no external forces act on it.
For our exercise, we use the momentum equation to describe this conservation:

• The total momentum before the collision is equal to the total momentum after the collision.
• This equation is written as: \[m_1 v_{1i} + m_2 v_{2i} = m_1 v_{1f} + m_2 v_{2f}\]

This equation allows us to connect the known values before the collision with the unknown velocities after the collision.

Velocity Calculation in Collisions

The key to solving any collision problem is determining the velocities of the objects involved after the collision. In our problem, we calculated the unknown final velocity of the lighter disk by rearranging the momentum equation and solving for the unknown quantity:
The steps included substituting known values into the equation and performing basic arithmetic operations until the unknown velocity was isolated. Here's a brief summary of how this worked:

• Substitute the known values: \[(0.090)(3.00) + (0.075)(-8.00) = (0.090)(-7.00) + (0.075)v_{2f}\]
• Simplify and solve the equation to isolate \(v_{2f}\).
• Solve: \[-0.330 = -0.630 + 0.075v_{2f}\] results in the final velocity \(v_{2f} = 4.00\,\text{m/s}\).

Mastering Physics Problem Solving

Physics problems, especially those involving collisions, often challenge students to apply theoretical knowledge in practical scenarios. Here are some helpful approaches:

• Always start by carefully identifying the known and unknown variables.
• Understand the physical principles involved—in this case, conservation of momentum.
• Write down the key equations and substitute the known values accurately.
• Perform arithmetic operations methodically and check units regularly.

By mastering these skills, one can tackle a wide array of physics problems with confidence and precision.