A cyclist approaches the bottom of a gradual hill at a speed of 14 m/s. The hill is 4.0 m high, and the cyclist estimates that she is going fast enough to coast up and over it without peddling. Ignoring air resistance and friction, find the speed at which the cyclist crests the hill.

initial kinetic energy >

potential energy change
at the top of the hill.

Solve for V. Mass cancels out

To find the speed at which the cyclist crests the hill, we can use the principle of conservation of mechanical energy. This principle states that the total mechanical energy of a system stays constant if the only forces acting on it are conservative forces, such as gravity.

Let's break down the problem into different parts. Initially, at the bottom of the hill, the cyclist has kinetic energy due to their speed. As they move up the hill, this kinetic energy is gradually converted into gravitational potential energy. At the top of the hill, all of the kinetic energy is transformed into potential energy.

The total mechanical energy, E, of the system remains constant, meaning the sum of the kinetic and potential energies at any point is always equal to the initial total mechanical energy.

Mathematically, we can express this as:

E = K + U,

where E is the total mechanical energy, K is the kinetic energy, and U is the gravitational potential energy.

Since the cyclist estimates coasting up and over the hill without pedaling, we can assume the cyclist does not input any work, and therefore, no energy is lost due to friction or air resistance. This allows us to ignore these factors.

At the bottom of the hill, the cyclist has kinetic energy (K1) given by:

K1 = 1/2 * m * v1^2

where m is the mass of the cyclist, and v1 is the initial speed (14 m/s).

At the top of the hill, the cyclist only has gravitational potential energy (U2) given by:

U2 = m * g * h2

where g is the acceleration due to gravity (approximately 9.8 m/s^2), and h2 is the height of the hill (4.0 m).

Since the total mechanical energy remains constant, we can equate the initial kinetic energy to the final potential energy:

K1 = U2.

1/2 * m * v1^2 = m * g * h2.

Simplifying the equation:

v1^2 = 2 * g * h2.

Plugging in the known values:

v1^2 = 2 * 9.8 m/s^2 * 4.0 m.

v1^2 = 78.4 m^2/s^2.

Taking the square root of both sides:

v1 = √(78.4) ≈ 8.84 m/s.

Therefore, the speed at which the cyclist crests the hill is approximately 8.84 m/s.