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A \(0.25\) -kg ball moving in the \(+x\) -direction at \(13 \mathrm{~m} / \mathrm{s}\) is hit by a bat. Its final velocity leaving the bat is \(19 \mathrm{~m} / \mathrm{s}\) in the \(x\) -direction. The bat acts on the ball for \(0.010 \mathrm{~s}\). Find the average force \(F\) exerted on the ball by the bat. The problem provides the time over which a required force acts, as well as enough information to compute the change in momentum. That suggests the impulse equation (i.e., Newton's Second Law). We have \(v_{i}=13 \mathrm{~m} / \mathrm{s}\) and \(v_{f}=-19 \mathrm{~m} / \mathrm{s}\). Taking the initial direction of motion as positive, the impulse equation is $$ \begin{aligned} F \Delta t &=m v_{f}-m v_{i} \\ F(0.010 \mathrm{~s}) &=(0.25 \mathrm{~kg})(-19 \mathrm{~m} / \mathrm{s})-(0.25 \mathrm{~kg})(13 \mathrm{~m} / \mathrm{s}) \end{aligned} $$ from which \(F=-0.80 \mathrm{kN}\).

Step by step solution

01

Understand Given Values

We have a ball with mass \( m = 0.25 \) kg. Its initial velocity \( v_i = 13 \) m/s is in the \(+x\)-direction, and its final velocity \( v_f = -19 \) m/s is in the \(-x\)-direction. The time interval \( \Delta t = 0.010 \) s during which the force was applied is given.

02

Identify the Formula

We need to use the impulse-momentum theorem, which states that the impulse applied to an object is equal to the change in its momentum: \[ F \Delta t = m(v_f - v_i) \] where \( F \) is the average force, \( \Delta t \) is the time duration, \( m \) is the mass, \( v_i \) is the initial velocity, and \( v_f \) is the final velocity.

03

Substitute Known Values into Formula

Substitute \( m = 0.25 \) kg, \( v_i = 13 \) m/s, \( v_f = -19 \) m/s, and \( \Delta t = 0.010 \) s into the impulse equation:\[ F(0.010) = 0.25(-19) - 0.25(13) \]

04

Calculate the Change in Momentum

First, calculate each momentum component:- Initial momentum = \( m \cdot v_i = 0.25 \cdot 13 = 3.25 \) kg·m/s- Final momentum = \( m \cdot v_f = 0.25 \cdot (-19) = -4.75 \) kg·m/s- Change in momentum = \( -4.75 - 3.25 = -8 \) kg·m/s

05

Solve for the Average Force

Using the impulse formula, we have:\[ F \cdot 0.010 = -8 \, \text{kg·m/s} \]Solve for \( F \):\[ F = \frac{-8}{0.010} = -800 \, \text{N} \] or \( -0.80 \) kN.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Average Force Calculation

In physics, calculating the average force is crucial when understanding how forces interact with objects over time. The average force exerted on an object can be determined if you know the time duration over which the force has been applied and how the object's momentum changes during this time.
The formula used is given by the impulse-momentum theorem, where the impulse provided during a collision or interaction is equal to the average force multiplied by the time period it acts: \[ F \Delta t = m(v_f - v_i) \]Here:

• \( F \) is the average force exerted.
• \( \Delta t \) is the time during which the force acts.
• \( m \) is the mass of the object.
• \( v_i \) and \( v_f \) are the initial and final velocities respectively.

The average force considers both the size of the force and the duration. In this example, a negative sign in the result indicates the force direction is opposite to the initial direction of motion.

Change in Momentum

The change in momentum of an object results from the difference between the initial and final momentum. Momentum, a product of mass and velocity, is a measure of the motion of an object.
When you have a situation where an external force is applied, momentum can change, even if the mass remains constant. This change is calculated by measuring the difference between the initial and final momentum:
\[ \text{Change in momentum} = m v_f - m v_i \]In this problem:

• Initial momentum is the product of initial velocity and mass: \( m \cdot v_i \).
• Final momentum is the product of final velocity and mass: \( m \cdot v_f \).
• The difference gives the change: \(-4.75 - 3.25 = -8\) kg·m/s.

This change indicates how the speed and direction of the ball were altered due to an external influence (the bat). A negative change reveals a reversal or decrease in motion.

Impulse Equation

The impulse equation is a fundamental law in mechanics. It directly links the force applied to an object with a corresponding change in momentum over a given time period.
Impulse (\( J \)) is defined as:

• \( J = F \Delta t \)

Where \( J \) is the impulse, \( F \) the average force, and \( \Delta t \) is the time duration the force is applied.
Using this formula, it becomes evident that to produce a certain change in momentum, an equivalent impulse, or force multiplied by time, is required:
\[ m(v_f - v_i) = F \Delta t \]This is incredibly useful, as it allows us to derive the force if the timing and momentum change are known. While similar-sized forces might seem intuitive, this theorem reveals that both dimension size and application duration are integral components in force calculations.