Question

To warm up for a match, a tennis player hits the 57.0-g ball vertically with her racket. If the ball is stationary just before it is hit and goes 5.50 m high, what impulse did she impart to it?

Short answer

The impulse is approximately 0.592 kg⋅m/s.

Step-by-step solution

Identify Known Values and Units

The mass of the tennis ball is given as 57.0 g, which needs to be converted to kilograms. Thus, \( m = 0.057 \) kg. The ball reaches a height of 5.50 m, so \( h = 5.50 \) m. The acceleration due to gravity is \( g = 9.81 \text{ m/s}^2 \).

Use Energy Conservation to Find Final Velocity

To find the velocity of the ball just after it is hit, apply conservation of energy. Initially, the kinetic energy (KE) is converted into gravitational potential energy (PE) at the peak height. Therefore, \( \frac{1}{2} m v^2 = mgh \), which simplifies to \( v = \sqrt{2gh} \). Substituting \( g = 9.81 \text{ m/s}^2 \) and \( h = 5.50 \text{ m} \), we find \( v = \sqrt{2 \times 9.81 \times 5.5} \approx 10.38 \text{ m/s} \).

Calculate the Impulse

The impulse imparted to the ball can be calculated using the change in momentum, \( J = \Delta p = m \Delta v \). Since the initial velocity \( v_i = 0 \) (the ball is stationary), \( \Delta v = v_f - v_i = 10.38 \text{ m/s} \). Thus, \( J = 0.057 \times 10.38 \approx 0.592 \text{ kg} \cdot \text{m/s} \).

Key concepts

Conservation of Energy

The principle of conservation of energy is a fundamental idea in physics. It states that energy cannot be created or destroyed, only transformed from one form to another. In the context of the tennis ball problem, initially, the ball has kinetic energy (due to its motion after being hit) and this energy is entirely transformed into gravitational potential energy at its highest point.
When the ball is hit, it gains kinetic energy, which is calculated using the formula \( KE = \frac{1}{2} mv^2 \). As the ball rises to a height of 5.50m, this kinetic energy is converted into potential energy, given by \( PE = mgh \), where \( h \) is the height.
Using the conservation of energy formula \( \frac{1}{2} mv^2 = mgh \), we can derive the velocity of the ball right after it's hit. Here, we see that the initial kinetic energy equals the potential energy at the peak height, allowing us to solve for velocity by rearranging the terms and substituting the known values.

Momentum

Momentum is the measure of the amount of motion a body possesses and is the product of its mass and velocity. It's a vector quantity, meaning it has both magnitude and direction.
In our tennis ball scenario, the concept of impulse is closely tied to momentum. Impulse is defined as the change in momentum, represented by the formula \( J = \Delta p \). For an object in linear motion like the tennis ball, this can be expressed as \( J = m \Delta v \), where \( \Delta v \) is the change in velocity of the ball.

• If the initial velocity is zero (as the ball is stationary before being hit), \( \Delta v \) simplifies to \( v_f \), where \( v_f \) is the final velocity.
• The impulse provided by the racket is responsible for this change in momentum, calculated as the mass of the ball multiplied by the velocity right after impact.

This relationship highlights how forces and actions imparted on objects lead to changes in their state of motion.

Kinematics

Kinematics deals with the motion of objects without considering the forces that cause this motion. In the tennis ball example, we are interested in the ball's motion after it leaves the racket.
When the player hits the ball, it accelerates upwards, reaching a maximum velocity computed from energy principles. As it ascends, the ball decelerates due to gravity until its velocity reaches zero at the peak. This is where we apply the kinematics equations to describe its motion:
For a body moving under constant acceleration, kinematic equations help us calculate key variables such as velocity and displacement. Here, the initial kinematics problem involved determining how high the ball would go, directly influencing how we understood its velocity after being struck.

• Even though forces like gravity act on the ball, in kinematics the focus is solely on the positional and temporal aspects, avoiding forces.
• Through its journey from the racket to its highest point, analyzing velocity and acceleration gives insights into the motion sequence.

The integration of kinematic concepts with energy and momentum helps provide a complete understanding of motion in physical systems.