Problem 11 - College Physics

Question

Baseball. A regulation 145 g baseball can be hit at speeds of 100 mph. If a line drive is hit essentially horizontally at this speed and is caught by a 65 \(\mathrm{kg}\) player who has leapt directly upward into the air, what horizontal speed (in \(\mathrm{cm} / \mathrm{s} )\) does he acquire by catching the ball?

Step-by-Step Solution

Answer

The player acquires a horizontal speed of approximately 9.95 cm/s.

Step 1: Convert Baseball Speed to SI Units

First, convert the speed of the baseball from miles per hour (mph) to meters per second (m/s). We know that 1 mile per hour is approximately 0.44704 meters per second. Thus, the speed of the baseball in m/s is calculated as follows: \[ 100 \text{ mph} \times 0.44704 \dfrac{\text{m}}{\text{s}} = 44.704 \dfrac{\text{m}}{\text{s}}. \]

Step 2: Momentum Before and After Collision

Before the collision, only the baseball has horizontal momentum. The momentum of the baseball is given by the product of its mass and velocity. The momentum of the player is initially zero since he is moving vertically.Using the formula for momentum, \[p = m \times v,\] calculate for the baseball:\[p_{\text{baseball}} = 0.145 \text{ kg} \times 44.704 \dfrac{\text{m}}{\text{s}}.\]

Step 3: Calculate Initial Baseball Momentum

Substitute the known values into the momentum formula:\[p_{\text{baseball}} = 0.145 \text{ kg} \times 44.704 \dfrac{\text{m}}{\text{s}} = 6.48108 \dfrac{\text{kg m}}{\text{s}}.\]

Step 4: Apply the Principle of Conservation of Momentum

According to the principle of conservation of momentum, the total momentum before the player catches the ball is equal to the total momentum after he catches it. If the player's speed after catching the ball is \(v_f\), the equation becomes:\[ p_{\text{total before}} = p_{\text{total after}} \]\[ 6.48108 \dfrac{\text{kg m}}{\text{s}} = (65 \text{ kg} + 0.145 \text{ kg}) \times v_f. \]

Step 5: Solve for the Player's Final Horizontal Speed

Now, solve for \(v_f\):\[ v_f = \frac{6.48108 \dfrac{\text{kg m}}{\text{s}}}{65.145 \text{ kg}} = 0.09946 \dfrac{\text{m}}{\text{s}}.\]

Step 6: Convert Final Speed to Desired Units

Finally, convert the player's final speed from meters per second to centimeters per second (cm/s), knowing that 1 m/s equals 100 cm/s:\[ 0.09946 \dfrac{\text{m}}{\text{s}} \times 100 \dfrac{\text{cm}}{\text{m}} = 9.946 \dfrac{\text{cm}}{\text{s}}.\]

Key Concepts

Momentum

Momentum is a key concept in physics that describes the quantity of motion an object possesses. It's calculated by multiplying an object's mass by its velocity. The formula is:

• \( p = m \times v \)

where \( p \) represents momentum, \( m \) stands for mass, and \( v \) denotes velocity.

In linear motion, both the mass and the direction of the velocity make up the momentum. This means that heavier or faster-moving objects have more momentum than lighter or slower ones.

When analyzing collisions, momentum allows us to understand how much motion is transferred between objects. In our baseball scenario, we looked at the momentum of the baseball before it hits the player's mitt. Calculating the initial momentum helps determine the player's speed once the collision occurs.

Unit Conversion

Unit conversion is essential when solving physics problems because it ensures consistency in calculations. Different units require conversion to work together in equations.

In our example, we needed to convert the baseball's speed from miles per hour (mph) to meters per second (m/s), a standard unit of speed in physics. This conversion works using the factor:

• 1 mph \( \approx 0.44704 \text{ m/s} \).

Another conversion occurs when we determine the player's final speed, this time from meters per second to centimeters per second. This conversion is simple because

• 1 m/s equals 100 cm/s.

Understanding and applying these conversions are crucial for getting accurate answers in physics problems.

Linear Motion

Linear motion refers to movement along a straight path. In our problem, the horizontal motion of the baseball and the player are examples of linear motion. When studying linear motion, physics considers factors such as velocity and acceleration, but in this exercise, we focus primarily on velocity.

The baseball moves with a certain horizontal velocity as it approaches the player. Initially, the player is not moving horizontally since he leapt vertically to catch the ball. Therefore, his horizontal velocity is zero.

When the ball is caught, some of the baseball's horizontal velocity transfers to the player, causing him to move horizontally as well, thus demonstrating the interaction of linear motion through collision.

Collision Analysis

Collision analysis involves understanding how objects interact when they collide. In physics, this often includes studying momentum before and after the collision to determine resultant movements.

The principle of conservation of momentum states that the total momentum of a closed system remains constant through a collision. This means that in the absence of external forces, the momentum before and after the collision will be equal.

In our exercise, we applied this principle, calculating the momentum of the baseball before it was caught by the player, then setting it equal to the combined momentum of the player and ball after the catch.

This approach enabled us to determine the horizontal speed that the player acquired, emphasizing how collision analysis helps solve practical, real-world physics problems.