Problem 98 - University Physics with Modern Physics

Question

Suppose you hold a small ball in contact with, and directly over, the center of a large ball. If you then drop the small ball a short time after dropping the large ball, the small ball rebounds with surprising speed. To show the extreme case, ignore air resistance and suppose the large ball makes an elastic collision with the floor and then rebounds to make an elastic collision with the still-descending small. Just before the collision between the two balls, the large ball is moving upward with velocity \(\vec{\boldsymbol{v}}\) and the small ball has velocity \(-\vec{\boldsymbol{v}}\) . (Do you see why? \()\) Assume the large ball has a much greater mass than the small ball. (a) What is the velocity of the small ball immediately after its collision with the large ball? (b) From the answer to part (a), what is the ratio of the small ball's rebound distance to the distance it fell before the collision?

Step-by-Step Solution

Answer

After collision, the small ball moves with velocity \(3\vec{\mathbf{v}}\) upward. Rebound distance is 9 times the fall distance.

Step 1: Understand Elastic Collision

In an elastic collision, both momentum and kinetic energy are conserved. The two balls, large and small, will collide elastically when they meet because, in the context of this exercise, we ignore air resistance or any other form of energy loss.

Step 2: Define the problem scenario

Since the small ball is initially moving downward with velocity \(-\vec{\mathbf{v}}\) and the large ball is moving upward with velocity \(\vec{\mathbf{v}}\), they collide when their paths meet. Given that the large ball has a much greater mass, it will mostly keep its velocity and impart a new velocity to the smaller ball.

Step 3: Apply the principle of Conservation of Momentum

For two objects with masses \(m_1\) and \(m_2\) (where \(m_1\gg m_2)\) and velocities \(v_1\) and \(v_2\), the momentum before and after collision must remain consistent. Assuming perfectly elastic conditions, the combined system's initial momentum, due to the large mass of the bigger ball, changes only the velocity of the small ball.

Step 4: Calculate Velocity of Small Ball After Collision

With the large ball hardly slowing down due to its mass, the velocity of the small ball reverses direction quickly. By applying elastic collision equations, we find: \[v' = 3\vec{\mathbf{v}}\] This is because the large ball initially moves at \(\vec{\mathbf{v}}\) and transfers a significantly greater magnitude to the small ball as it rebounds.

Step 5: Determine the ratio of rebound to descent

The small ball, now moving upwards at 3 times its initial descent speed as \(3\vec{\mathbf{v}}\), will ascend much further than it fell. Hence, the rebound distance will be 9 times the descent distance, as kinetic energy influences its potential journey upwards.

Key Concepts

Conservation of Momentum

In the world of physics, the principle of conservation of momentum is essential when analyzing collisions. It tells us that the total momentum of a system remains constant, provided there are no external forces acting on it.

For our exercise, this principle is quite straightforward. The momentum just before the collision between the large and small ball should match the momentum right afterward.

Here's the key: - The momentum of an object is calculated as product of its mass and its velocity. - If we denote the mass and initial velocity of the large ball as \(m_1\) and \(v_1 = \vec{\mathbf{v}}\), and the small ball as \(m_2\) and \(v_2 = -\vec{\mathbf{v}}\), the conservation of momentum states: \[m_1v_1 + m_2v_2 = m_1v_1' + m_2v_2'\] where \(v_1'\) and \(v_2'\) are the velocities of the large and small balls after the collision.

Given that \(m_1 \gg m_2\), the large ball's velocity remains largely unaffected, meaning all the conservation shifts the small ball's velocity, making it essential to factor in for further calculations.

Kinetic Energy

Understanding kinetic energy is crucial, especially in cases involving elastic collisions, like the one described here. Kinetic energy is the energy that an object possesses due to its motion, and it's given by the formula: \[KE = \frac{1}{2}mv^2\] where \(m\) is mass and \(v\) is velocity.

In elastic collisions, both kinetic energy and momentum are conserved. This means the total kinetic energy before the collision will equal the total after the collision. To break it down: - The large ball carries a significant portion of the system's initial kinetic energy due to both its mass and velocity.- The small ball, even though it has lesser mass, undergoes a remarkable increase in velocity after the collision.

This results in the small ball rebounding with heightened energy, which is why it speeds away so dramatically post-collision. This conservation is mirrored in its kinetic energy, as the small ball's rebound speed skyrockets.

Velocity Calculations

Let's dive into how we calculate the velocity of the small ball after it collides elastically with the large ball. Using our understanding of momentum conservation and elastic collision, we know the small ball’s velocity gains a multiplier due to the interaction.

The important points to remember include: - Before collision, the velocity of the small ball is \(-\vec{\mathbf{v}}\) and of large ball is \(\vec{\mathbf{v}}\).- Post-collision, due to conservation and the relative immobility of the larger ball, the velocity of the small ball becomes \(v' = 3\vec{\mathbf{v}}\). - This is because the smaller ball receives additional momentum imparted by the moving large ball.

Such velocity calculations are prime examples of the delicate balance maintained in physics between energy transfer and momentum conservation. After collision, the small ball’s velocity being triple its descent speed illustrates how collisions can fundamentally alter motion dynamics.