A grandfather pulls his grandson, whose mass is 20kg and who is sitting on a swing with ropes of length 2.50m, backward and releases him from rest. The speed of the grandson at the bottom of the swinging motion is 3.0 m/s. What is the angle (in degrees measured relative to the vertical) from which he is released?

Try using conssrvation of energy.

Hint: The change in elevation for an angle A from vertical is
L (1 - cos A)
where L is the rope length.

To find the angle from which the grandson is released, we can use the conservation of energy principle. At the bottom of the swing, the kinetic energy can be calculated using the formula:

KE = 1/2 * m * v^2

where KE is the kinetic energy, m is the mass of the grandson, and v is the speed at the bottom.

The potential energy at the highest point is equal to the gravitational potential energy at the lowest point:

PE = m * g * h

Where PE is the potential energy, m is the mass, g is the acceleration due to gravity (approximately 9.8 m/s^2), and h is the height.

Since the rope forms a right-angled triangle with the vertical, we can use trigonometry to relate the height to the length of the swing:

h = L * (1 - cosθ)

Where L is the length of the swing and θ is the angle from which the grandson was released.

Equating the potential energy and kinetic energy:

m * g * h = 1/2 * m * v^2

Substituting the expression for h, we get:

m * g * (L * (1 - cosθ)) = 1/2 * m * v^2

Canceling out the mass and simplifying:

g * (L * (1 - cosθ)) = 1/2 * v^2

Simplifying further:

(1 - cosθ) = (1/2) * v^2 / (g * L)

cosθ = 1 - (1/2) * v^2 / (g * L)

θ = arccos(1 - (1/2) * v^2 / (g * L))

Now plugging in the given values:

θ = arccos(1 - (1/2) * (3.0)^2 / (9.8 * 2.50))

Calculating this value gives us:

θ ≈ 46.43 degrees

Therefore, the angle from which the grandson was released is approximately 46.43 degrees relative to the vertical.

To find the angle from which the grandson is released, we can analyze the conservation of mechanical energy.

First, let's calculate the potential energy of the grandson when he is released. At the highest point of the swing, all the potential energy is converted into kinetic energy at the bottom of the swing.

The potential energy at the highest point is given by:

Potential Energy = m * g * h

Where:
m = mass of the grandson = 20 kg
g = acceleration due to gravity = 9.8 m/s²
h = height of the swing at the highest point

Since the swing is released from rest, the potential energy at the highest point is equal to the kinetic energy at the bottom point:

Potential Energy = Kinetic Energy

m * g * h = (1/2) * m * v²

Where:
v = speed of the grandson at the bottom of the swing = 3.0 m/s

Simplifying the equation:

g * h = (1/2) * v²

Solving for h:

h = (1/2) * v² / g

Substituting the given values:

h = (1/2) * (3.0 m/s)² / 9.8 m/s²

Now, we can use the law of cosines to find the angle from which the grandson is released. The law of cosines states that:

c² = a² + b² - 2 * a * b * cos(C)

Where:
a = length of the rope = 2.50 m
b = height of the swing at the highest point = h
c = total length of the rope when the swing is at the lowest point (2.50 m)

Substituting the values:

(2.50 m)² = (2.50 m)² + (h)² - 2 * (2.50 m) * (h) * cos(C)

Simplifying the equation:

0 = (2.50 m)² + (h)² - 2 * (2.50 m) * (h) * cos(C) - (2.50 m)²

0 = (h)² - 2 * (2.50 m) * (h) * cos(C)

Now, we can substitute the value of h:

0 = [(1/2) * (3.0 m/s)² / 9.8 m/s²]² - 2 * (2.50 m) * [(1/2) * (3.0 m/s)² / 9.8 m/s²] * cos(C)

Solving this equation will give us the value of cos(C). Taking the inverse cosine (cos⁻¹) of that value will give us the angle C in radians. Finally, to convert the angle from radians to degrees, we multiply it by (180/π).

By following this process, you will be able to calculate the angle (in degrees) from which the grandson is released.