91Ó°ÊÓ

The concept of impulse is crucial when analyzing situations involving forces and motion over short periods. Impulse measures the impact of a force applied over time. Calculating impulse involves the formula:

• Impulse (\( J \)) = Force (\( F \)) \( \times \) Time (\( t \))

In this exercise, we have forces provided in two perpendicular directions, the x- and y-components. For the x-component, we use the force of \(-380 \text{ N}\) and a time of \(3.00 \text{ ms} = 3 \times 10^{-3} \text{ s}\). Thus, the impulse in the x-direction, \(J_x\), is calculated as:

• \( J_x = -380 \text{ N} \times 3 \times 10^{-3} \text{ s} = -1.14 \text{ Ns}\)

In the y-direction, the force is \(110 \text{ N}\), maintaining the same time period. Hence, the impulse, \(J_y\), is:

• \( J_y = 110 \text{ N} \times 3 \times 10^{-3} \text{ s} = 0.33 \text{ Ns}\)

Impulse provides a direct measure of how a force changes the momentum of an object. This concept ties directly into momentum through Newton's laws of motion.

Determining the final velocity requires understanding the relationship between impulse and change in momentum. Momentum is the product of mass and velocity, and changes in momentum relate to the impulse:

• Change in momentum = Impulse

Given that mass can be expressed through weight and gravitational acceleration, i.e., mass (\( m \)) = weight / \( 9.8 \text{ m/s}^2 \), the mass of the tennis ball is \( 0.056 \text{ kg} (\text{from}\, 0.560 \text{ N})\).For the x-component, starting with an initial velocity \( V_{xi} = 20 \text{ m/s}\):

• Final velocity: \( V_{xf} = V_{xi} + \frac{J_x}{m} = 20 \text{ m/s} - \frac{1.14 \text{ Ns}}{0.056 \text{ kg}} = -20 \text{ m/s}\)

For the y-component, initial velocity is \( V_{yi} = -4 \text{ m/s}\):

• Final velocity: \( V_{yf} = V_{yi} + \frac{J_y}{m} = -4 \text{ m/s} + \frac{0.33 \text{ Ns}}{0.056 \text{ kg}} = 2 \text{ m/s}\)

The final velocities reflect how external force changes the trajectory and speed of the ball in both directions.

In physics, motion is often defined by forces, and the relationship between these is outlined by Newton's second law of motion. This core law explains how a net force acting on an object results in acceleration, affecting its velocity and position over time. Here, the ball experiences a net force during its contact with the racket, influencing its speed.The exercise shows that the force into components states how differing directions of force can distinctly change motion. The understanding of motion affected by force is:

• Force (\( F \)) in x- and y-direction affects motion based on \( F = ma \) (where \( a \) is acceleration).

The constant forces provided alter the ball’s speed and direction:

• In x-direction: \(-380 \text{ N}\) pulls back, \( \text{updating direction}\).
• In y-direction: \(110 \text{ N}\) directs upwards, affecting elevation.

The application and effect of net external forces on an object's motion can be seen practically in sports like tennis, for predictive motion analysis.

Understanding component vectors is fundamental in physics when dealing with multidimensional motion. Vectors have both magnitude and direction, making them applicable in describing forces and velocities in distinct directions. In this problem: The tennis ball's velocity and forces applied are split into x and y components:

• Initial Velocity Vector: \((20 \hat{i}) - (4 \hat{j})\,\text{m/s}\)
• Net External Forces:\(-380 \hat{i} + 110 \hat{j}\,\text{N}\)

The importance of components lies in:

• Having clarity on how each separate force impacts direction.
• Ability to handle vector addition or subtraction, thus solving for final states like velocity.

Grasping the separation of vectors into components simplifies the solution of many physics problems, ensuring precise calculation and understanding of multi-directional forces.

Newton's Laws of Motion are the foundation for understanding mechanics. They describe the relationship between an object's motion and the forces acting upon it. Primarily:- **First Law**: An object in motion stays in motion unless acted upon by an external force.- **Second Law**: The net force on an object is equal to the mass of the object multiplied by its acceleration (\( F = ma \)).- **Third Law**: For every action, there is an equal and opposite reaction.In this exercise, the Second Law is pivotal. It relates how the constant net force during the ball's contact period results in acceleration, directly influencing the change in velocity:

• Provides a predictive understanding of force implications on the tennis ball's motion.
• Gives insight into explaining how speed decreases or increases based on direction and magnitude of force.

Using these laws simplifies analyzing everyday phenomena such as playing tennis, where forces cater to motion patterns and provide a deeper insight into the mechanics involved.