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?

Verified step by step guidance

1. Identify the initial and final velocities of the tennis ball. Initially, the ball is stationary, so its initial velocity (u) is 0 m/s. When the ball reaches its maximum height, its final velocity (v) is also 0 m/s.
2. Use the kinematic equation for vertical motion to find the initial velocity just after being hit. The equation is v^2 = u^2 + 2as, where 'a' is the acceleration due to gravity (approximately -9.81 m/s^2, negative as it acts downward) and 's' is the displacement (5.50 m).
3. Calculate the change in momentum, or impulse, imparted to the ball. Impulse can be calculated using the formula Impulse = m \Delta v, where 'm' is the mass of the ball and \Delta v is the change in velocity from just after being hit to when it stops at the peak.
4. Convert the mass of the tennis ball from grams to kilograms to maintain consistency in units for the impulse calculation.
5. Substitute the values into the impulse formula to find the impulse imparted to the ball. Remember that the impulse will be equal in magnitude but opposite in direction to the change in momentum caused by gravity.

Key Concepts

Impulse

Impulse is defined as the change in momentum of an object when a force is applied over a period of time. It is calculated as the product of the average force and the time duration over which the force acts. In this context, the impulse imparted to the tennis ball can be determined by the change in its momentum as it moves from rest to its peak height.

Kinetic and Potential Energy

Kinetic energy is the energy of an object due to its motion, while potential energy is the energy stored in an object due to its position in a gravitational field. When the tennis ball is hit and rises to a height of 5.50 m, its kinetic energy is converted into gravitational potential energy, which can be calculated using the formula PE = mgh, where m is mass, g is the acceleration due to gravity, and h is height.

Conservation of Energy

The principle of conservation of energy states that energy cannot be created or destroyed, only transformed from one form to another. In this scenario, the mechanical energy of the tennis ball is conserved as it transitions from kinetic energy (when hit) to potential energy (at the peak height). This relationship allows us to calculate the initial velocity of the ball and subsequently the impulse imparted by the player.