A 46.7-g golf ball is driven from the tee with an initial speed of 55.5 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 5.85 m below its highest point?
a. Ek = 0.5m*V^2 = 0.5*0.0467*0^2 = 0.
Note: The velocity is 0 at the max. ht.
b. V^2 = Vo^2 + 2g*h
V^2 = 0 + 19.6*(25.9-5.85) = 391
V = 19.8 m/s.
To solve this problem, we need to use the principles of conservation of energy. The total mechanical energy of the golf ball is conserved as it rises and then falls back down.
(a) To find the kinetic energy of the ball at its highest point, we need to determine its potential energy at that height and subtract it from the total energy.
The potential energy (PE) of an object at a certain height is given by the equation:
PE = m * g * h
Where m is the mass of the golf ball, g is the acceleration due to gravity, and h is the height.
Using the given values:
m = 46.7 g = 0.0467 kg (converting grams to kilograms)
g = 9.8 m/s^2 (approximate value for gravity on Earth)
h = 25.9 m
PE = 0.0467 kg * 9.8 m/s^2 * 25.9 m
PE = 11.9 J (rounded to one decimal place)
The total mechanical energy (E) of the golf ball is the sum of its kinetic energy (KE) and potential energy (PE) at any point in its motion. So, at the highest point, when the golf ball reaches its maximum height, the kinetic energy is equal to the total energy minus the potential energy.
E = KE + PE
Since air resistance is neglected, the total mechanical energy is constant.
Therefore, at the highest point, the kinetic energy is:
KE = E - PE
KE = E - 11.9 J
(b) To find the speed of the ball when it is 5.85 m below its highest point, we can use the concept of conservation of mechanical energy. The total mechanical energy at this point will be the sum of the kinetic energy and the potential energy at that height.
E = KE + PE
Since air resistance is neglected, the total mechanical energy is constant.
At the highest point, we determined that the kinetic energy is E - 11.9 J.
Now, let's calculate the potential energy (PE) when the ball is 5.85 m below its highest point.
PE = m * g * h
Using the given values:
m = 0.0467 kg
g = 9.8 m/s^2
h = (25.9 m - 5.85 m) = 20.05 m
PE = 0.0467 kg * 9.8 m/s^2 * 20.05 m
PE = 9.1 J (rounded to one decimal place)
The kinetic energy (KE) at 5.85 m below the highest point is:
KE = E - PE
KE = E - 9.1 J
Note: Since we don't know the specific value of the total mechanical energy (E), we cannot determine the actual kinetic energy or speed at that specific point without additional information. However, we can express the kinetic energy as a function of the total mechanical energy and solve for the speed.
To determine the kinetic energy of the golf ball at its highest point, we need to use the principle of conservation of energy. According to this principle, the total mechanical energy of the ball remains constant throughout its motion, assuming there is no external work or energy dissipation due to factors like air resistance.
(a) At its highest point, the golf ball has reached its maximum potential energy and no longer has any kinetic energy. Therefore, we can equate the initial kinetic energy to the potential energy at the highest point.
The initial kinetic energy of the ball can be calculated using the formula:
KE = (1/2) * mass * velocity^2
Where:
- KE is the kinetic energy,
- mass is the mass of the golf ball,
- velocity is the initial speed of the ball.
Substituting the given values:
mass = 46.7 g = 0.0467 kg (since 1 g = 0.001 kg)
velocity = 55.5 m/s
KE = (1/2) * 0.0467 kg * (55.5 m/s)^2
(b) To find the speed of the ball when it is 5.85 m below its highest point, we can again use the principle of conservation of energy. We need to determine the potential energy at this height and equate it to the sum of kinetic and potential energies at its highest point.
The potential energy of the ball at a height h is given by the formula:
PE = mass * g * h
Where:
- PE is the potential energy,
- mass is the mass of the golf ball,
- g is the acceleration due to gravity, approximately 9.8 m/s^2,
- h is the height.
The potential energy at the golf ball's highest point is equal to the kinetic energy at that point, which is zero, as explained earlier.
So, we can write the equation:
PE at height h = KE at highest point
mass * g * h = (1/2) * mass * velocity^2
Substituting the given values:
mass = 46.7 g = 0.0467 kg (since 1 g = 0.001 kg)
g = 9.8 m/s^2
h = 5.85 m
0.0467 kg * 9.8 m/s^2 * 5.85 m = (1/2) * 0.0467 kg * velocity^2
Now, we can solve this equation to find the value of velocity, which will be the speed of the ball when it is 5.85 m below its highest point.