Problem 20 - University Physics with Modern Physics

Question

You are standing on a sheet of ice that covers the football stadium parking lot in Buffalo; there is negligible friction between your feet and the ice. A friend throws you a 0.400 -kg ball that is traveling horizontally at 10.0 \(\mathrm{m} / \mathrm{s}\) . Your mass is 70.0 \(\mathrm{kg}\) . (a) If you catch the ball, with what speed do you and the ball move afterward? (b) If the ball hits you and bounces off your chest, so afterward it is moving horizontally at 8.0 \(\mathrm{m} / \mathrm{s}\) in the opposite direction, what is your speed after the collision?

Step-by-Step Solution

Answer

(a) 0.0568 m/s; (b) 0.103 m/s.

Step 1: Understand the Problem

We're given a situation where we need to determine the speed of you and a ball after a collision on a frictionless surface. We need to solve two parts: (a) when you catch the ball, and (b) when the ball bounces back.

Step 2: Conservation of Momentum for (a)

For part (a), we apply conservation of momentum. Initially, the total momentum is due to the ball: \[ p_{initial} = m_{ball} \times v_{ball} = 0.400 \times 10.0 = 4.0 \text{ kg m/s} \]After catching, you and the ball move together, so:\[ p_{final} = (m_{you} + m_{ball}) \times v_{final} \]Equating initial and final momentum gives:\[ 4.0 = (70.0 + 0.400) \times v_{final} \]

Step 3: Solve for Final Velocity in (a)

Solve for \(v_{final}\):\[ 4.0 = 70.4 \times v_{final} \]\[ v_{final} = \frac{4.0}{70.4} \approx 0.0568 \text{ m/s} \]

Step 4: Conservation of Momentum for (b)

For part (b), again apply conservation of momentum. Initially:\[ p_{initial} = 4.0 \text{ kg m/s} \] The ball bounces back and a new momentum for the ball is:\[ p'_{ball} = m_{ball} \times (-8.0) = 0.400 \times (-8.0) = -3.2 \text{ kg m/s} \] After collision, the total momentum is:\[ p_{final} = m_{you} \times v_{you} + p'_{ball} \]Equating initial and final momentum:\[ 4.0 = 70.0 \times v_{you} - 3.2 \]

Step 5: Solve for Your Velocity in (b)

Rearrange and solve for \(v_{you}\):\[ 4.0 + 3.2 = 70.0 \times v_{you} \]\[ 7.2 = 70.0 \times v_{you} \]\[ v_{you} = \frac{7.2}{70.0} \approx 0.103 \text{ m/s} \]

Key Concepts

Collision Physics

In collision physics, we are interested in what happens when two objects hit each other. Collisions can happen in various ways, but they mostly adhere to the same basic principles. These principles involve changes in the velocity and direction of the involved objects and are governed mainly by the conservation of momentum. In any collision, we consider:

• The initial velocities of the objects before they collide.
• The final velocities of the objects after the collision.
• Whether the collision is elastic or inelastic, which will tell us whether kinetic energy is conserved along with momentum.

In this particular exercise on a frictionless surface, you are catching or being hit by a ball. Whether the ball is caught or bounces off your chest affects the final outcome, but fundamentally, both scenarios are analyzed using the laws of momentum conservation. Understanding collisions helps us predict the motion of objects post-collision, which is critically important in fields ranging from traffic safety to sports physics.

Frictionless Surface

A frictionless surface is an ideal concept used often in physics problems to simplify the calculations. In reality, most surfaces have some degree of friction, but by assuming frictionlessness, we can better understand the core interactions between objects without the additional complexity presented by frictional forces. When there is no friction:

• Moving objects do not lose energy to the surface they are on.
• There is no resistance to the change in motion or direction.
• The only forces you need to consider are those directly acting on the objects (such as the force from a collision).

In our exercise, the lack of friction means that when the ball is thrown and caught, or hits and bounces off, the only calculations we need to make are those involving the momentum of the ball and your momentum. This greatly simplifies our understanding and allows us to focus purely on the interaction of the ball with you.

Momentum Calculation

Momentum is one of the core concepts in physics and is defined as the product of an object's mass and its velocity. It is a vector quantity, possessing both magnitude and direction, and is vital when calculating the outcomes of collisions. The formula for momentum is:\[p = m imes v\]Where:

• \( p \) is momentum.
• \( m \) is the mass of the object.
• \( v \) is the velocity of the object.

During collisions, the principle of conservation of momentum tells us that the total momentum before the collision equals the total momentum after. This allows us to set up equations that relate the initial and final velocities of the colliding objects after considering their masses.In the exercise, you initially calculated the momentum of the ball and then considered how it changed through two different scenarios: when you catch the ball and when it bounces off. These calculations highlight how momentum principles allow us to predict the resultant velocities after collisions.