Tom is on a swing at a park. In order to swing higher, Sam applies a certain force to the swing. If the speed of the swing is 6 m/s at the lowest point, how high will the swing rise above the lowest point?

The swing will rise 1.84 meters above the lowest point.

Explanation:

Suppose that we have a pendulum.

The total energy of the pendulum can be written as:

E = U + K

where U is the potential energy:

U = m*g*h

m is the mass, g is the gravitational acceleration, and h is the height.

and K is the kinetic energy:

K = (m/2)*v^2

where v is the velocity.

When the swing is on the higher point, its velocity is zero, thus, at that point it only has potential energy.

When the swing is in the lowest point, its height is zero (we define h = 0 in that point) thus the potential energy is zero, and it only has kinetic energy.

And because of the conservation of energy, both of these energies must be equal.

The kinetic energy in the lowest point is:

K = (m/2)*(6m/s)^2

the potential energy in the highest point is:

U = m*(9.8m/s^2)*h

These two quantities are equal, then:

(m/2)*(6m/s)^2 = m*(9.8m/s^2)*h

Now we can solve this for h, the maximum height.

now we can remove the mass in both sides to get:

(1/2)*(36 m^2/s^2) = (9.8m/s^2)*h

Solving for h, we get:

(1/2)*(36 m^2/s^2)/(9.8m/s^2) = h = 1.84 m

The swing will rise 1.84 meters above the lowest point.

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