A steel ball has a mass of 4.0 kg and rolls along a smooth, level surface at 62m/s. What is its kinetic energy. At first the ball was at rest on the surface. A force acted on it through a distance of 22 m to give it the speed of 62 m/s. What was the magnitude of the force?

Since it is rolling you must consider both the kinetic energy of both translation (forward motion) and rotation. The rotational kinetic energy is (1/2) I w^2 where I = (2/5) M R^2 for a solid sphere, and w = V/R is the angular velocity.

Put it all together and you find that the kinetic energy is (7/10)M V^2

Divide that by the distance of 22m over which the force acted, and you get the average force.

To find the kinetic energy of the steel ball, we can use the formula:

Kinetic Energy = 1/2 * mass * velocity^2

Given that the mass of the steel ball is 4.0 kg and its velocity is 62 m/s, we can plug these values into the formula to calculate the kinetic energy:

Kinetic Energy = 1/2 * 4.0 kg * (62 m/s)^2

Kinetic Energy = 0.5 * 4.0 kg * 3844 m^2/s^2

Kinetic Energy = 3844 joules

Therefore, the kinetic energy of the steel ball is 3844 joules.

Now, let's move on to calculating the magnitude of the force.

We know that work is done on the steel ball to accelerate it from rest to a speed of 62 m/s. The work done is given by the formula:

Work = Force * Distance

Since we are given the distance traveled by the ball is 22 m, and the final velocity is 62 m/s, we can rearrange the formula to solve for force:

Force = Work / Distance

To find the work done on the ball, we can use the formula:

Work = Change in Kinetic Energy

The change in kinetic energy is the final kinetic energy minus the initial kinetic energy. Initially, the ball was at rest, so the initial kinetic energy is zero. Therefore:

Change in Kinetic Energy = Final Kinetic Energy - Initial Kinetic Energy
= Kinetic Energy - 0
= Kinetic Energy

Substituting the value we found earlier for the kinetic energy:

Change in Kinetic Energy = 3844 joules

Finally, we can calculate the magnitude of the force:

Force = Change in Kinetic Energy / Distance
= 3844 joules / 22 m
= 174.73 newtons (rounded to two decimal places)

Therefore, the magnitude of the force is approximately 174.73 newtons.