The string in the Figure is L = 135.0 cm long and the distance d to the fixed peg P is 94.5 cm. When the ball is released from rest in the position shown, it will swing along the dashed arc. How fast will it be going when it reaches the lowest point in its swing? How fast will it be going when it reaches the highest point in its swing?

To determine how fast the ball will be going when it reaches the lowest and highest points in its swing, we can use the concepts of conservation of energy and mechanical energy.

To find the speed at the lowest point, we can use the principle of conservation of mechanical energy. At the highest point, all the potential energy of the ball is converted into kinetic energy. Similarly, at the lowest point, the kinetic energy is at its maximum and potential energy is at its minimum.

1. Finding the speed at the lowest point:
To find the speed at the lowest point, we can equate the potential energy at the lowest point to the kinetic energy at the lowest point.

Potential energy at the lowest point = mgh
Kinetic energy at the lowest point = (1/2)mv^2

Where:
m is the mass of the ball
g is the acceleration due to gravity (approximately 9.8 m/s^2)
h is the height of the lowest point above the ground

We can find h using the given information:
Length of the string (L) = 135.0 cm = 1.35 m
Distance to the fixed peg (d) = 94.5 cm = 0.945 m

h = L - d

After finding h, we can equate the potential energy to the kinetic energy and solve for v, which represents the speed at the lowest point.

2. Finding the speed at the highest point:
To find the speed at the highest point, we can equate the kinetic energy at the highest point to the potential energy at the lowest point.

Potential energy at the lowest point = mgh
Kinetic energy at the highest point = (1/2)mv^2

We already know the potential energy at the lowest point, and the kinetic energy at the highest point will be equal to the potential energy at the lowest point. Therefore, we can equate the two and solve for v, which represents the speed at the highest point.

By going through these steps, we can calculate the speed at the lowest point and the speed at the highest point of the swing.

To calculate the speed at the lowest and highest points in the swing, we can use the principles of conservation of energy.

Step 1: Find the potential energy at the starting position.
- The potential energy (PE) at the starting position is given by PE = mgh, where m is the mass of the ball, g is the acceleration due to gravity, and h is the height of the ball above the lowest point of the swing.
- In this case, since the ball is released from rest, the height h is equal to the length of the string L.
- Assuming the mass of the ball is negligible compared to other forces, we can ignore it.
- Therefore, the potential energy at the starting position is PE_start = mgh = 0 (since the ball is released).

Step 2: Find the potential energy at the lowest point.
- At the lowest point, the ball will have its maximum kinetic energy and zero potential energy.
- Therefore, the potential energy at the lowest point is PE_lowest = 0.

Step 3: Find the potential energy at the highest point.
- At the highest point, the ball will have its maximum potential energy and zero kinetic energy.
- Therefore, the potential energy at the highest point is given by PE_highest = mgh, where h is the difference between the length of the string L and the distance d to the fixed peg P (h = L - d).

Step 4: Calculate the speed at the lowest point.
- The total mechanical energy of the system (consisting of potential energy and kinetic energy) is conserved.
- Therefore, the total mechanical energy at the starting position (E_start) is equal to the total mechanical energy at the lowest point (E_lowest).
- The total mechanical energy is given by the sum of potential energy (PE) and kinetic energy (KE): E = PE + KE.
- Since the potential energy at the starting position is 0, the mechanical energy at the starting position is E_start = 0.
- At the lowest point, all the potential energy is converted into kinetic energy, so the potential energy at the lowest point is 0.
- Therefore, the mechanical energy at the lowest point is also 0: E_lowest = 0.
- The kinetic energy (KE) is given by KE = (1/2)mv^2, where m is the mass of the ball and v is the speed of the ball.
- Therefore, at the lowest point, KE_lowest = (1/2)mv^2 = 0.
- Solving for v, we find that the speed at the lowest point is v_lowest = 0.

Step 5: Calculate the speed at the highest point.
- Using the same conservation of energy principle, the mechanical energy at the starting position (E_start) is equal to the mechanical energy at the highest point (E_highest).
- The mechanical energy at the highest point is given by E_highest = PE_highest + KE_highest.
- Since the potential energy at the starting position is 0, the mechanical energy at the starting position is E_start = 0.
- At the highest point, all the kinetic energy is converted into potential energy, so the kinetic energy at the highest point is 0.
- Therefore, the mechanical energy at the highest point is also 0: E_highest = 0.
- Solving for KE_highest = (1/2)mv^2, we find that the kinetic energy at the highest point is KE_highest = 0.
- At the highest point, the potential energy is given by PE_highest = mgh, where h = L - d.
- Therefore, PE_highest = mgh = mg(L - d).

Step 6: Calculate the speed at the highest point.
- Since the mechanical energy at the highest point is 0, we can set the potential energy equal to the negative kinetic energy: PE_highest = -KE_highest.
- This gives us mg(L - d) = -(1/2)mv_highest^2.
- Solving for v_highest, we find that the speed at the highest point is given by v_highest = sqrt(2g(L - d)).

In summary:
- The speed at the lowest point in the swing is v_lowest = 0.
- The speed at the highest point in the swing is v_highest = sqrt(2g(L - d)).