A 47.8-g golf ball is driven from the tee with an initial speed of 53.6 m/s and rises to a height of 32.6 m.

(a) Neglect air resistance and determine the kinetic energy of the ball at its highest point.
(b) What is its speed when it is 7.24 m below its highest point?

Oh, so the golf ball is going on a cosmic adventure! Let's see how it's doing.

(a) Well, at the highest point, the ball is about to take a little break from its speedy journey. Since it's reaching its peak, we can say that all the initial kinetic energy has been converted into potential energy. So, the kinetic energy at the highest point is zero. It's like the golf ball decided to take a moment to catch its breath and enjoy the view.

(b) Now, when the ball is 7.24 m below its highest point, it's starting to dive down a bit. But worry not, we can still calculate its speed! To do this, we can use the conservation of energy principle. At this point, the potential energy of the ball would be converted back into kinetic energy. So, subtract the potential energy at 7.24 m below the highest point from the initial potential energy and you'll get the kinetic energy. Once you have that, you can use it to find the speed by using the good old kinetic energy formula.

Now, let's get calculating and give that adventurous golf ball some answers!

To solve this problem, we can use the principles of conservation of energy. The total mechanical energy of an object is the sum of its kinetic energy (KE) and potential energy (PE) at any given point. Since air resistance is neglected, the total mechanical energy of the golf ball remains constant throughout its motion.

(a) To determine the kinetic energy of the ball at its highest point, we need to find its potential energy first. The potential energy (PE) of an object of mass m at a height h is given by the formula:

PE = m * g * h

where g is the acceleration due to gravity (approximately 9.8 m/s^2).

Given:
mass of the golf ball (m) = 47.8 g = 0.0478 kg
height (h) = 32.6 m

Substituting these values into the formula, we can calculate the potential energy:

PE = 0.0478 kg * 9.8 m/s^2 * 32.6 m
= 15.24 J

Since the total mechanical energy of the ball is conserved, the potential energy at its highest point is equal to the kinetic energy at that point. Thus, the kinetic energy (KE) at the highest point is also 15.24 J.

(b) To determine the speed of the ball when it is 7.24 m below its highest point, we can use the principle of conservation of energy again.

Let v be the velocity of the ball when it is 7.24 m below its highest point. The total mechanical energy at this point is equal to the sum of kinetic energy and potential energy:

KE + PE = m * g * h + 0.5 * m * v^2

We already know the potential energy at this point (PE = 15.24 J). We need to find the kinetic energy at this point to solve for v.

To find the kinetic energy, we can use the formula:

KE = 0.5 * m * v^2

Substituting the known values:

15.24 J + 0.5 * 0.0478 kg * v^2 = 15.24 J

We can solve this equation to find the value of v.

To answer these questions, we need to understand the concepts of kinetic energy and potential energy and how they relate to each other.

(a) Kinetic energy is the energy of an object due to its motion. It can be calculated using the formula:

KE = (1/2)mv^2

where KE represents kinetic energy, m is the mass of the object, and v is the velocity of the object.

At its highest point, the golf ball's velocity is momentarily zero. This means that its kinetic energy is also zero, as there is no motion. However, we can calculate the kinetic energy just before it reaches the highest point.

Given:
Mass of the golf ball, m = 47.8 g = 0.0478 kg
Initial velocity, v = 53.6 m/s

Using the formula for kinetic energy, we can calculate:

KE = (1/2)mv^2
= (1/2)(0.0478 kg)(53.6 m/s)^2
≈ 64.843 J

Therefore, the kinetic energy of the golf ball just before reaching its highest point is approximately 64.843 Joules.

(b) To find the speed at a specific height below the highest point, we can use the conservation of energy principle. The total mechanical energy (sum of kinetic energy and potential energy) remains constant in the absence of external forces like air resistance.

In this case, the potential energy is given by:

PE = mgh

where PE represents potential energy, m is the mass of the object, g is acceleration due to gravity (approximately 9.8 m/s^2), and h is the height.

Given:
Height, h = 7.24 m

To find the speed at this height, we need to calculate the total mechanical energy (E) at the highest point and use it to determine the kinetic energy (KE) at the height of interest.

At the highest point, all the initial kinetic energy is converted into potential energy:

E = KE + PE = PE

Using the given height, we can calculate the potential energy at the highest point:

PE = mgh
= (0.0478 kg)(9.8 m/s^2)(32.6 m)
≈ 15.863 J

Now, we can find the velocity (v) at the given height by equating the potential energy at the highest point to the sum of kinetic and potential energies at the specific height:

E = KE + PE

KE = E - PE

Using the formula for kinetic energy and the calculated potential energy at the given height, we can calculate the speed:

KE = (1/2)mv^2

(1/2)mv^2 = E - PE

(1/2)(0.0478 kg)v^2 = (64.843 J) - (15.863 J)

(0.0239 kg)v^2 = 48.98 J

v^2 = (48.98 J) / (0.0239 kg)

v = √(48.98 J / 0.0239 kg)

v ≈ 36.37 m/s

Therefore, the speed of the golf ball when it is 7.24 m below its highest point is approximately 36.37 m/s.