A 45.7-g golf ball is driven from the tee with an initial speed of 46.0 m/s and rises to a height of 25.9 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 6.82 m below its highest point?

Kinetic Energy at any point= 1/2 m 46^2-mgh

put in h, and solve.

for b, h= 25.6-6.82

To calculate the kinetic energy of the golf ball at its highest point and determine its speed at a given position, we can utilize the principles of conservation of mechanical energy.

(a) To find the kinetic energy at the highest point, we need to first determine the potential energy at the highest point and then subtract it from the initial kinetic energy.

The potential energy of an object at a given height is given by the formula:

PE = m * g * h

where PE is the potential energy, m is the mass of the object, g is the acceleration due to gravity (approximately 9.8 m/s² near the surface of the Earth), and h is the height.

In this case, the mass of the golf ball is 45.7 g, which can be converted to kilograms by dividing by 1000:

m = 45.7 g / 1000 = 0.0457 kg

The height of the ball is 25.9 m.

PE = 0.0457 kg * 9.8 m/s² * 25.9 m = 11.99 J

Next, we subtract the potential energy from the initial kinetic energy to obtain the kinetic energy at the highest point.

Initial kinetic energy given: 46.0 m/s

KE = (1/2) * m * v^2

where KE is the kinetic energy and v is the velocity.

KE = (1/2) * 0.0457 kg * (46.0 m/s)^2

KE = 47.386 J

Finally, we subtract the potential energy:

Kinetic energy at highest point = KE - PE = 47.386 J - 11.99 J ≈ 35.40 J

(b) To find the speed of the golf ball when it is 6.82 m below its highest point, we simply need to calculate the total mechanical energy of the ball at that position and then use it to find the velocity.

The total mechanical energy, which remains constant in the absence of external forces such as air resistance, is the sum of kinetic energy and potential energy:

Total mechanical energy = KE + PE

Using the values determined earlier, the potential energy at the given position can be calculated:

PE = m * g * h

PE = 0.0457 kg * 9.8 m/s² * 6.82 m ≈ 3.03 J

Now, subtract the potential energy from the total mechanical energy:

Total mechanical energy = kinetic energy + potential energy

Total mechanical energy = 35.40 J + 3.03 J ≈ 38.43 J

To find the speed, we use the formula for kinetic energy:

(1/2) * m * v^2 = Total mechanical energy

(1/2) * 0.0457 kg * v^2 = 38.43 J

v^2 = (38.43 J) / (0.0457 kg * (1/2))

v^2 ≈ 1683.51 m²/s²

v ≈ √(1683.51 m²/s²)

v ≈ 41.02 m/s

Therefore, the speed of the golf ball when it is 6.82 m below its highest point is approximately 41.02 m/s.