Problem 44 - College Physics

Question

A bat strikes a 0.145 kg baseball. Just before impact, the ball is traveling horizontally to the right at \(50.0 \mathrm{m} / \mathrm{s},\) and it leaves the bat traveling to the left at an angle of \(30^{\circ}\) above horizontal with a speed of 65.0 \(\mathrm{m} / \mathrm{s}\) . (a) What are the horizontal and vertical components of the impulse the bat imparts to the ball? (b) If the ball and bat are in contact for 1.75 \(\mathrm{ms}\) , find the horizontal and vertical components of the average force on the ball.

Step-by-Step Solution

Answer

The horizontal impulse is \(-22.559 \text{ kg} \cdot \text{m/s}\), vertical impulse is \(4.7125 \text{ kg} \cdot \text{m/s}\). Horizontal force is \(-12,891.14 \text{ N}\), vertical force is \(2,692.86 \text{ N}\).

Step 1: Determine Initial and Final Velocities

First, define the initial and final velocities of the ball. The initial velocity \( v_{i} \) is given as 50.0 m/s horizontally to the right. The final velocity \( v_{f} \) has a magnitude of 65.0 m/s at an angle of \( 30^{\circ} \) above the horizontal to the left. We need to calculate both horizontal \( v_{fx} \) and vertical \( v_{fy} \) components of the final velocity. Using trigonometry, these components can be calculated as follows: \( v_{fx} = 65.0 \times \cos(30^{\circ}) \) (which will be negative, to the left) and \( v_{fy} = 65.0 \times \sin(30^{\circ}) \).

Step 2: Calculate Horizontal Impulse

Impulse \( J \) is defined as the change in momentum. Calculate the horizontal component of the impulse using the formula: \( J_x = m(v_{fx} - v_i) \). Substitute the values: \( m = 0.145 \text{ kg}, v_i = 50.0 \text{ m/s}, \) and \( v_{fx} = -65.0 \times \cos(30^{\circ}) \text{ m/s} \). Thus, \( J_x = 0.145(-65.0 \times \cos(30^{\circ}) - 50.0) \).

Step 3: Calculate Vertical Impulse

Similarly, calculate the vertical component of the impulse using \( J_y = m(v_{fy} - 0) \) because the initial vertical velocity is zero. So, \( J_y = 0.145 \times 65.0 \times \sin(30^{\circ}) \).

Step 4: Determine Average Forces

Average force is the impulse divided by the time in contact. To find average horizontal force \( F_x \), use \( F_x = \frac{J_x}{t} \) where \( t = 1.75 \times 10^{-3} \text{ s} \). For the average vertical force \( F_y \), use \( F_y = \frac{J_y}{t} \). Apply the values calculated for \( J_x \) and \( J_y \) from the previous steps.

Key Concepts

Horizontal and Vertical Components

Understanding horizontal and vertical components is crucial when analyzing motion, especially in physics problems involving angles and directions. When a force or velocity acts at an angle, it can be broken down into horizontal and vertical components using trigonometry. This makes the analysis simpler.

When the bat strikes the baseball, the ball's final velocity is not straightforward because it travels at an angle. To fully understand its motion, we must divide the resultant velocity into its horizontal and vertical parts.

• The horizontal component (\(v_{fx}\) ) is calculated using cosine: \(v_{fx} = v_f \times \cos(\theta)\). Since the ball moves to the left, this value will be negative.
• The vertical component (\(v_{fy}\)) comes from the sine: \(v_{fy} = v_f \times \sin(\theta)\).

By analyzing these components, the changes in momentum, hence the impulse imparted by the bat, can be clearly identified.

Trigonometry in Physics

In physics, trigonometry helps us resolve a vector, such as velocity, into its component parts. This is essential when dealing with any motion that isn't along a single axis.

In our baseball problem, the final velocity makes a \(30^{\circ}\) angle above the horizontal, requiring the use of trigonometric functions.

Here's how it helps:

• Trigonometric functions like sine and cosine help you find components of a vector. Cosine deals with the adjacent (horizontal) component, and sine with the opposite (vertical) component of an angle.
• This breakdown of vectors using trigonometry is useful in simplifying complex motion into manageable parts.

Applying these principles helps bridge the angle with its Cartesian components, making calculations on the changing momentum accurate and straightforward.

Average Force Calculations

Average force gives insight into how strong the hit was over the period of contact. It's essentially the impulse divided by the time the forces were in action.To calculate average force:

• First find the impulse, which is the change in momentum. For horizontal impulse \(J_x\), it's \( m(v_{fx} - v_i) \).
• Vertical impulse \(J_y\) relies on the final vertical velocity since the initial one is zero, \( m(v_{fy}) \).
• Finally, compute the average force by dividing each impulse component by the contact time \(t\).

So, \( F_x = \frac{J_x}{t} \) and \( F_y = \frac{J_y}{t} \).

This reveals the continuous average force needed to cause that change in momentum, showing the strength and direction of the interaction.