A 47.0 g golf ball is driven from the tee with an initial speed of 54.0 m/s and rises to a height of 23.8 m.
(a) Neglect air resistance and determine the kinetic energy of the ball at its highest point.
J
(b) What is its speed when it is 10.0 m below its highest point?
m/s
(a) Subtract the potential energy at 23.8 m from the initial kinetic energy.
(KE at highest pt.)
= (initial KE) - M g H
= (0.047 kg)*(54.0 m/s)^2/2
- (0.047)(9.8)(23.80 =
(b) 10.0 m below highest point is 13.8 m above tee elevation. Initial KE is decreased by M*g*13.8 m = 6.4 Joules at that elevation. Use that KE to get the corresponding speed
This formula is not very clear to me
(a) Well, the golf ball must be pretty elated reaching such new heights! To find its kinetic energy at the highest point, we need to consider its initial kinetic energy and the change in potential energy. Now imagine the ball on a roller coaster ride: as it goes up, potential energy increases, and as it goes down, kinetic energy increases. At the highest point, potential energy is at its maximum, so the kinetic energy must be at its minimum. Therefore, at the highest point, the kinetic energy is zero joules.
(b) Ah, the golf ball is on a downward spiral! To find its speed 10.0 m below its highest point, we can use the conservation of energy. Since we're neglecting air resistance, we know that total energy is conserved. So, the potential energy at the highest point (with respect to the starting point) must be equal to the sum of the kinetic and potential energy at 10.0 m below the highest point.
Since the question only provides the height at the highest point and the mass of the golf ball, it doesn't directly give us the potential energy. But hey, let's see what we can do with what we have!
First, let's calculate the potential energy at the highest point:
Potential energy = mass * gravity * height
Potential energy = 47.0 g * 9.8 m/s² * 23.8 m
Now, we can find the speed at 10.0 m below the highest point. Let's say the speed is "v."
Potential energy at 10.0 m below highest point = kinetic energy + potential energy
Mass * gravity * (23.8 m - 10.0 m) = (1/2) * mass * v² + mass * gravity * 10.0 m
Now, let's solve for "v." Take it away, math!
In the meantime, I'll be over here, working on my golf swing. Just remember, hitting the ball too hard might make it clown around and take flight like a circus performer!
(a) To determine the kinetic energy of the golf ball at its highest point, we first need to calculate the gravitational potential energy when the ball is at its highest point.
The gravitational potential energy can be calculated using the formula:
PE = m * g * h
where m is the mass of the ball (47.0 g = 0.047 kg), g is the acceleration due to gravity (9.8 m/s^2), and h is the height (23.8 m).
PE = 0.047 kg * 9.8 m/s^2 * 23.8 m
≈ 10.8434 J
At its highest point, all the initial kinetic energy (K.E.) has been converted to potential energy (P.E.). Therefore, the kinetic energy at the highest point is equal to the initial kinetic energy of the ball.
K.E. = 10.8434 J
So, the kinetic energy of the ball at its highest point is approximately 10.8434 J.
(b) To determine the speed of the golf ball when it is 10.0 m below its highest point, we need to use the law of conservation of energy.
At its highest point, the total mechanical energy (potential energy + kinetic energy) is constant and equal to the initial total mechanical energy.
Initial total mechanical energy = K.E. + P.E.
Let's calculate the initial total mechanical energy:
Initial total mechanical energy = K.E. + P.E.
Initial total mechanical energy = 0.5 * m * v^2 + m * g * h
where v is the initial velocity (54.0 m/s), m is the mass (47.0 g = 0.047 kg), g is the acceleration due to gravity (9.8 m/s^2), and h is the height (23.8 m).
Initial total mechanical energy = 0.5 * 0.047 kg * (54.0 m/s)^2 + 0.047 kg * 9.8 m/s^2 * 23.8 m
Initial total mechanical energy ≈ 7.36632 J
Now, we can set up the equation for the final total mechanical energy at a height of 10.0 m below its highest point:
Final total mechanical energy = K.E. + P.E.
Final total mechanical energy = 0.5 * m * v'^2 + m * g * h'
where v' is the final velocity (unknown), and h' is the height (10.0 m).
Final total mechanical energy = 0.5 * 0.047 kg * v'^2 + 0.047 kg * 9.8 m/s^2 * 10.0 m
Since the total mechanical energy is conserved, we can equate the initial and final total mechanical energies:
Initial total mechanical energy = Final total mechanical energy
7.36632 J = 0.5 * 0.047 kg * v'^2 + 0.047 kg * 9.8 m/s^2 * 10.0 m
Now we can solve for v':
7.36632 J = 0.5 * 0.047 kg * v'^2 + 0.047 kg * 9.8 m/s^2 * 10.0 m
Rearranging the equation:
0.5 * 0.047 kg * v'^2 = 7.36632 J - 0.047 kg * 9.8 m/s^2 * 10.0 m
Simplifying:
0.0235 kg * v'^2 = 7.36632 J - 4.586 J
0.0235 kg * v'^2 = 2.78032 J
Dividing both sides by 0.0235 kg:
v'^2 ≈ 118.31064 m^2/s^2
Taking the square root:
v' ≈ √(118.31064 m^2/s^2)
v' ≈ 10.88 m/s
So, the speed of the golf ball when it is 10.0 m below its highest point is approximately 10.88 m/s.
To solve this problem, we can use conservation of energy. At the highest point, the kinetic energy of the ball will be converted completely into potential energy.
(a) To determine the kinetic energy at the highest point, we first need to calculate the potential energy. The potential energy of an object is given by the formula:
Potential Energy = mass * gravitational acceleration * height
Plugging in the values:
Potential Energy = (47.0 g) * (9.8 m/s^2) * (23.8 m)
Remember to convert the mass from grams to kilograms by dividing by 1000:
Potential Energy = (47.0 g / 1000 kg/g) * (9.8 m/s^2) * (23.8 m)
Potential Energy = 111.044 J
Since the potential energy at the highest point is equal to the total energy (kinetic energy + potential energy), the kinetic energy at the highest point is:
Kinetic Energy = Total Energy - Potential Energy
However, in this case, the total energy is equal to the kinetic energy at the initial point because we are neglecting air resistance. So, the answer is:
Kinetic Energy = (47.0 g) * (54.0 m/s)^2 /2
Remember to convert the mass from grams to kilograms by dividing by 1000:
Kinetic Energy = (47.0 g / 1000 kg/g) * (54.0 m/s)^2 /2
Kinetic Energy = 66.123 J
Therefore, the kinetic energy of the golf ball at its highest point is 66.123 J.
(b) To find the speed when the ball is 10.0 m below its highest point, we need to determine the potential energy at that point and convert it back to kinetic energy. We can use the same formula for potential energy as before, and substitute the height with 10.0 m:
Potential Energy = (47.0 g / 1000 kg/g) * (9.8 m/s^2) * (10.0 m)
Potential Energy = 4.558 J
Using the formula for total mechanical energy, we have:
Total Energy = Kinetic Energy + Potential Energy
Since the total energy is conserved, we can write:
Initial Kinetic Energy = Final Kinetic Energy + Final Potential Energy
Solving for the final kinetic energy:
Final Kinetic Energy = Initial Kinetic Energy - Final Potential Energy
Final Kinetic Energy = (47.0 g) * (54.0 m/s)^2 /2 - 4.558 J
Remember to convert the mass from grams to kilograms by dividing by 1000:
Final Kinetic Energy = (47.0 g / 1000 kg/g) * (54.0 m/s)^2 /2 - 4.558 J
Final Kinetic Energy = 38.765 J
To find the speed, we use the formula for kinetic energy:
Kinetic Energy = (1/2) * mass * velocity^2
Solving for velocity:
velocity = sqrt(2 * (Final Kinetic Energy) / mass)
velocity = sqrt(2 * (38.765 J) / (47.0 g / 1000 kg/g))
velocity = sqrt(0.829 m^2/s^2)
velocity = 0.91 m/s
Therefore, the speed of the golf ball when it is 10.0 m below its highest point is 0.91 m/s.