A 48.9-g golf ball is driven from the tee with an initial speed of 45.4 m/s and rises to a height of 29.9 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 5.62 m below its highest point?
Oh, a golf ball with some height and speed! Let's dive into it.
(a) To find the kinetic energy at the highest point, we need to know that kinetic energy is the energy an object possesses due to its motion. At the highest point, the golf ball momentarily stops moving vertically before it starts coming back down. So, its kinetic energy would be zero. Poor ball, it's taking a breather up there!
(b) The speed of the ball when it is 5.62 m below its highest point can be calculated using the concept of conservation of mechanical energy. The total mechanical energy of the ball, which includes both kinetic and potential energy, remains constant throughout its motion. So we can equate the initial mechanical energy to the mechanical energy at any lower point.
Since we're given the initial speed and height, we can use the equation:
1/2 mv^2 + mgh = 1/2 mv₀^2 + mgh₀
where m is the mass of the golf ball, v is its velocity, v₀ is the initial velocity, g is the acceleration due to gravity, h is the height, and h₀ is the initial height.
Substituting the given values and solving for v:
1/2 * 48.9g * v^2 + 48.9g * 5.62m = 1/2 * 48.9g * 45.4m/s^2 + 48.9g * 29.9m
After simplifying and solving, you'll find the speed when the ball is 5.62 m below its highest point.
But remember, my friend, this calculation neglects air resistance. In reality, golf balls might behave differently due to air drag. So, keep this in mind while teeing off!
To solve this problem, we can use the principle of conservation of mechanical energy, which states that the sum of kinetic and potential energies of an object remains constant if no external forces are acting on it.
(a) Let's start with determining the kinetic energy of the golf ball at its highest point. At the highest point, the potential energy (PE) is maximized, while the kinetic energy (KE) is minimized, as the ball momentarily comes to rest. Therefore, the kinetic energy at the highest point is equal to zero.
(b) To find the speed of the golf ball when it is 5.62 m below its highest point, we can use the conservation of mechanical energy.
The initial kinetic energy (KEi) of the ball is given by the formula:
KEi = (1/2) * m * v^2
where m is the mass of the ball and v is the initial velocity. Therefore, substituting the given values:
KEi = (1/2) * 48.9 g * (45.4 m/s)^2
To convert the mass from grams to kilograms, divide by 1000:
m = 48.9 g / 1000 = 0.0489 kg
Substituting the values, we get:
KEi = (1/2) * 0.0489 kg * (45.4 m/s)^2
Simplifying this equation gives us the initial kinetic energy:
KEi = 49.78 J
Now, let's find the potential energy at 5.62 m below its highest point. The potential energy (PE) is given by the formula:
PE = m * g * h
where m is the mass, g is the acceleration due to gravity (9.8 m/s^2), and h is the height. Substituting the given values:
PE = 0.0489 kg * 9.8 m/s^2 * 5.62 m
Simplifying this equation gives us the potential energy at 5.62 m below its highest point:
PE = 2.70 J
Since the sum of kinetic and potential energies is constant, we can find the speed (v) at 5.62 m below its highest point using the equation:
KEf + PEf = KEi + PEi
where KEf is the final kinetic energy at 5.62 m below its highest point and PEf is the final potential energy at 5.62 m below its highest point. Substituting the values:
0 + PEf = 49.78 J + 2.70 J
Simplifying this equation gives us the potential energy at 5.62 m below its highest point:
PEf = 52.48 J
Now, to find the final kinetic energy (KEf), subtract the final potential energy (PEf) from the initial kinetic energy (KEi):
KEf = KEi - PEf
Substituting the values:
KEf = 49.78 J - 52.48 J
Simplifying this equation gives us the final kinetic energy at 5.62 m below its highest point:
KEf = -2.70 J
Since kinetic energy cannot be negative, this implies that the golf ball does not reach the same speed 5.62 m below its highest point as it has at the initial speed of 45.4 m/s.
To answer part (a), we need to find the kinetic energy of the golf ball at its highest point, given its initial speed and neglecting air resistance.
The kinetic energy of an object is given by the formula:
KE = (1/2) * m * v^2
Where:
KE is the kinetic energy
m is the mass of the object
v is the velocity of the object
In this case, we are given:
m = 48.9 g = 0.0489 kg (convert grams to kilograms)
v = 45.4 m/s
Substituting these values into the formula, we can calculate the kinetic energy at the highest point:
KE = (1/2) * 0.0489 kg * (45.4 m/s)^2
Solving this equation will give us the answer to part (a).
To answer part (b), we need to find the speed of the golf ball when it is 5.62 m below its highest point.
To do this, we will use the principle of conservation of energy. At its highest point, all of the initial kinetic energy of the ball will be converted into potential energy. Thus, at any height below that, the sum of the potential and kinetic energy will be constant.
The potential energy at any point can be calculated using 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/s^2)
h is the height above the reference point
In this case, we need to find the speed of the ball when it is 5.62 m below its highest point. So, the height above the reference point is (29.9 m - 5.62 m).
First, we calculate the potential energy at the highest point using the given mass and height:
PE_highest = m * g * 29.9 m
Then, we calculate the potential energy at the point 5.62 m below the highest point:
PE_below = m * g * (29.9 m - 5.62 m)
Since the sum of potential and kinetic energy is constant, we can write:
PE_highest + KE_highest = PE_below + KE_below
Since we have already calculated the potential energy at the highest point (PE_highest), and the kinetic energy at the highest point can also be calculated using the formula from part (a), we can rearrange the equation to solve for the kinetic energy at the specific height below the highest point (KE_below).
Once we have the value of KE_below, we can use the formula for kinetic energy to calculate the speed at that point.
I hope this explanation helps you understand how to approach and solve the problem!