Two children, m1 = 50 kg and m2 = 55 kg, and a plastic sled of mass 35 kg start at rest and slide for 5.0 s down a hill. The force of gravity pulling the children and sled down the hill is 470 N but the snow provides a frictional force of 325 N. At the bottom of the hill, the children and sled encounter a frozen pond (frictionless surface). Upon sliding onto the frozen pond, the children and sled continue moving, without any additional force, across the middle to the other side.

If the pond has a circumference of 163 m, to the nearest tenth of a second, how long did it take the children and sled to travel to the other side of the pond?

14.2

To determine how long it took for the children and sled to travel to the other side of the pond, we can calculate their initial velocity as they slid down the hill. Once we have the initial velocity, we can then use it to calculate the time it takes to cover the circumference of the pond.

First, let's calculate the net force acting on the children and sled as they slide down the hill. The net force is the difference between the force of gravity and the force of friction:

Net force = Force of gravity - Force of friction
Net force = 470 N - 325 N
Net force = 145 N

Next, we can use the net force to calculate the acceleration of the children and sled down the hill. The acceleration can be found using Newton's second law:

Net force = mass * acceleration

Rearranging the equation, we get:

Acceleration = Net force / mass
Acceleration = 145 N / (50 kg + 55 kg + 35 kg)
Acceleration ≈ 1.05 m/s²

Now, we can calculate the initial velocity of the children and sled as they reach the bottom of the hill. Since they start from rest, the initial velocity will be 0 m/s. Their final velocity can be calculated using the equation:

Final velocity = Initial velocity + (Acceleration * Time)

Rearranging the equation, we get:

Time = (Final velocity - Initial velocity) / Acceleration

Since the final velocity is the velocity at the bottom of the hill, we can calculate it using the kinematic equation:

Final velocity² = Initial velocity² + (2 * Acceleration * Distance)

Substituting the known values, we get:

0 = Initial velocity² + (2 * 1.05 m/s² * Distance)
Initial velocity² = -2 * 1.05 m/s² * Distance
Initial velocity ≈ √(-2 * 1.05 m/s² * Distance)

The distance traveled down the hill can be found using the kinematic equation:

Distance = (1/2) * Acceleration * Time²

Substituting the known values, we get:

Distance = (1/2) * 1.05 m/s² * (5.0 s)²
Distance ≈ 13.1 m

Plugging the calculated distance back into the equation for the initial velocity, we get:

Initial velocity ≈ √(-2 * 1.05 m/s² * 13.1 m)
Initial velocity ≈ 10.98 m/s

Now that we have the initial velocity, we can calculate the time it takes for the children and sled to travel the circumference of the pond. The time can be found using the equation:

Time = Distance / Velocity

Substituting the known values, we get:

Time = 163 m / 10.98 m/s
Time ≈ 14.8 s

Therefore, to the nearest tenth of a second, it took the children and sled approximately 14.8 seconds to travel to the other side of the pond.