Problem 69 - University Physics with Modern Physics

Question

Just before it is struck by a racket, a tennis ball weighing 0.560 \(\mathrm{N}\) has a velocity of \((20.0 \mathrm{m} / \mathrm{s}) \hat{\imath}-(4.0 \mathrm{m} / \mathrm{s}) \hat{\boldsymbol{J}}\) . During the 3.00 \(\mathrm{ms}\) that the racket and ball are in contact, the net force on the ball is constant and equal to \(-(380 \mathrm{N}) \hat{\boldsymbol{\imath}}+(110 \mathrm{N}) \hat{\boldsymbol{J}}\) . (a) What are the \(x\) - and \(y\) -components of the impulse of the net force applied to the ball? (b) What are the \(x\) - and \(y\) -components of the final velocity of the ball?

Step-by-Step Solution

Answer

(a) Impulse components: \(-1.14 \mathrm{Ns}\) and \(0.33 \mathrm{Ns}\). (b) Final velocity components: \(0.02 \mathrm{m/s}\) and \(1.78 \mathrm{m/s}\).

Step 1: Understand Impulse

Impulse \( \vec{J} \) is the change in momentum of the object and is given by the product of force \( \vec{F} \) and the time interval \( \Delta t \) during which the force acts. The formula is \( \vec{J} = \vec{F} \cdot \Delta t \).

Step 2: Calculate Impulse Components

Given \( \vec{F} = -(380 \mathrm{N}) \hat{\boldsymbol{i}} + (110 \mathrm{N}) \hat{\boldsymbol{j}} \) and \( \Delta t = 3.00 \mathrm{ms} = 3.00 \times 10^{-3} \mathrm{s} \), the impulse components are:\[ J_x = F_x \cdot \Delta t = -380 \mathrm{N} \times 3.00 \times 10^{-3} \mathrm{s} = -1.14 \mathrm{Ns} \]\[ J_y = F_y \cdot \Delta t = 110 \mathrm{N} \times 3.00 \times 10^{-3} \mathrm{s} = 0.33 \mathrm{Ns} \]

Step 3: Initial Velocity Components

The initial velocity of the ball is \( \vec{v}_i = (20.0 \mathrm{m/s}) \hat{\boldsymbol{i}} - (4.0 \mathrm{m/s}) \hat{\boldsymbol{j}} \) with components:\[ v_{ix} = 20.0 \mathrm{m/s} \]\[ v_{iy} = -4.0 \mathrm{m/s} \]

Step 4: Determine Mass of the Ball

The weight \( W = 0.560 \mathrm{N} \) is given by \( W = m \cdot g \). Thus, the mass \( m \) of the ball is:\[ m = \frac{W}{g} = \frac{0.560 \mathrm{N}}{9.8 \mathrm{m/s}^2} \approx 0.0571 \mathrm{kg} \]

Step 5: Apply Impulse-Momentum Theorem

The impulse-momentum theorem states that \( \vec{J} = m (\vec{v}_f - \vec{v}_i) \), where \( \vec{v}_f \) is the final velocity. Solving for \( \vec{v}_f \), we have: \[ m \cdot \vec{v}_f = m \cdot \vec{v}_i + \vec{J} \] \[ v_{fx} = v_{ix} + \frac{J_x}{m} \] \[ v_{fy} = v_{iy} + \frac{J_y}{m} \]

Step 6: Calculate Final Velocity Components

Substitute to find the components:\[ v_{fx} = 20.0 \mathrm{m/s} + \frac{-1.14 \mathrm{Ns}}{0.0571 \mathrm{kg}} = 20.0 \mathrm{m/s} - 19.98 \mathrm{m/s} = 0.02 \mathrm{m/s} \]\[ v_{fy} = -4.0 \mathrm{m/s} + \frac{0.33 \mathrm{Ns}}{0.0571 \mathrm{kg}} = -4.0 \mathrm{m/s} + 5.78 \mathrm{m/s} = 1.78 \mathrm{m/s} \]

Key Concepts

Newton's Second Law

One of the fundamental principles of physics is Newton's Second Law of Motion. It states that the net force acting on an object is equal to the product of its mass and acceleration. In equation form, this is written as \( F = m imes a \). This law explains how the velocity of an object changes when subjected to an external force.
For instance, if you know the mass of a tennis ball and the forces applied to it, you can determine its acceleration. For our sphere, the force components are given: \(-380 \mathrm{N}\) in the \(\hat{\boldsymbol{i}}\) direction and \(110 \mathrm{N}\) in the \(\hat{\boldsymbol{j}}\) direction. The fact that the force is constant simplifies our calculations.
By knowing the mass, which we find using the weight and gravitational pull, you can apply Newton’s Second Law to determine how the motion of the ball changes when it interacts with external forces like the racket.

Impulse-Momentum Theorem

The Impulse-Momentum Theorem is a key concept in understanding how forces affect motion over time. Impulse \( \vec{J} \) is defined as the product of the force \( \vec{F} \) and the time period \( \Delta t \) over which the force acts, or \( \vec{J} = \vec{F} \cdot \Delta t \). This results in a change in momentum of an object and it shows how powerful forces are applied over time.
In our example, the force is applied for \(3.00 \mathrm{ms}\) and we can calculate the impulse components using the given force and time. Once we have the impulse, we can apply the Impulse-Momentum Theorem: \( \vec{J} = m(\vec{v}_f - \vec{v}_i) \). This enables you to calculate the change in velocity of the ball, helping you identify its final velocity components after the interaction.

Kinematics

Kinematics explores the motion of objects without considering the forces that produce this movement. It describes motion in terms of displacement, velocity, and acceleration.
In our problem, the initial velocity components of the tennis ball were \( (20.0 \mathrm{m/s}) \hat{\boldsymbol{i}} \) and \(- (4.0 \mathrm{m/s}) \hat{\boldsymbol{j}} \). With kinematics, we can further predict the motion of the ball after forces are applied.
Using the Impulse-Momentum theorem alongside Kinematics, the relationship between initial velocity, impulse, and mass can help you determine the new velocity after the racket impacts the ball. This understanding is essential in solving both components of velocity effectively.

Vector Components

When dealing with physics problems, forces and motion often occur in multiple dimensions. This requires the use of vector components, allowing us to break down vectors into perpendicular components.
In the tennis ball exercise, both the force exerted and the ball’s velocity are given in vector form: for the force \( -(380 \mathrm{N}) \hat{\boldsymbol{i}} + (110 \mathrm{N}) \hat{\boldsymbol{j}} \) and for velocity \( (20.0 \mathrm{m/s}) \hat{\boldsymbol{i}} - (4.0 \mathrm{m/s}) \hat{\boldsymbol{j}} \).
By resolving these into components, you can handle each direction independently, using tools like the Impulse-Momentum theorem to explore how each direction behaves when forces are applied. This process simplifies solving complex physics problems and enhances understanding of motion in two or more dimensions.