You and a friend are going to a river, where they have a rope swing. Your friend, who has a mass of 60 kg, hops on the end of the rope. You tug horizontally on your friend with a force of 300 N, pulling them back until you can't move them any more. Then you let go and watch them swing out over the river. When the rope swing is at it's farthest point your friend lets go, eventually splashing into the river. If the rope is 30 m long, what total distance in m did your friend travel from her starting point to where she let go?

Details and assumptions
The acceleration of gravity is −9.8 m/s^2.
Ignore air resistance.

To calculate the total distance your friend traveled from her starting point to where she let go, we need to break down the problem into different stages.

1. When you are pulling your friend:
You are pulling your friend horizontally with a force of 300 N. Since you reach a point where you can't move them anymore, we can assume that the force you apply is equal to the maximum static friction force between your friend and the ground, which prevents them from being pulled further.

The formula for static friction force is:
Friction force (Ff) = coefficient of static friction (μs) * normal force (Fn)

Assuming a coefficient of static friction of 1 between your friend's shoes and the ground, we can calculate the normal force as the product of your friend's mass (m) and the acceleration due to gravity (g).

Normal force (Fn) = m * g

Fn = 60 kg * (-9.8 m/s^2)
Fn = -588 N (since gravity acts downwards)

Since the applied force is equal to the maximum static friction force, we have:
300 N = -588 N

This is not possible since the applied force is greater than the maximum static friction force. However, we will assume that the force you apply is enough to overcome static friction for simplicity. Keep in mind that this is an idealized assumption.

2. Motion after letting go:
Once your friend is released, they swing on the rope. At the highest point, the tension in the rope is equal to the weight of your friend. At this point, the tension in the rope provides the centripetal force necessary to keep your friend moving in a circular path.

The formula for centripetal force is:
Centripetal force (Fc) = (mass (m) * velocity^2 (v^2)) / radius (r)

At the highest point, the centripetal force is equal to the weight of your friend:
Fc = m * g

Substituting the known values, we get:
m * g = m * v^2 / r

We can rearrange the equation to solve for the velocity (v):
v^2 = g * r

v^2 = -9.8 m/s^2 * 30 m
v^2 = -294 m^2/s^2

Since velocity cannot be negative, we only consider the positive square root:
v = √(-294 m^2/s^2)
v = 17.14 m/s

3. Total distance traveled by your friend:
Now we can calculate the total distance traveled by your friend from her starting point to where she let go. This distance consists of two parts:

a) Horizontal distance while being pulled: This is the distance covered while you were pulling your friend until you reached the maximum static friction force. The force you applied is in the horizontal direction, so this distance is the product of the applied force and the displacement in the horizontal direction.

Displacement (d) = applied force (F) / friction force (Ff)

d = 300 N / -588 N
d = -0.51 m (negative sign indicates the opposite direction of the applied force)

b) Arc length while swinging: This is the distance covered along the circular path when your friend lets go. The arc length can be calculated by multiplying the angle covered (in radians) by the radius of the circle.

The angle covered can be found using the relationship between the velocity (v) and the radius (r):
v = angular velocity (ω) * r

ω = v / r

Since the rope swing covers a semi-circular path, the angle covered is π radians (180 degrees).

Arc length (s) = angle (θ) * radius (r)

s = π * 30 m
s = 94.25 m

The total distance traveled is the sum of the horizontal distance and the arc length:
Total distance = |d| + s (taking the magnitude of the negative displacement)

Total distance = 0.51 m + 94.25 m
Total distance = 94.76 m

Therefore, your friend traveled a total distance of 94.76 meters from her starting point to where she let go.