A cyclist approaches the bottom of a gradual hill at a speed of 14 m/s. The hill is 4.9 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.

Vf = square root of (V0^2 + 2g (h0 - h1)

Vf = square root of (14 m/s)^2 + 2(9.81)(4.9)

Vf = square root of (196m^2/s^2 + 96.138

Vf = square root of (292.138)

Vf = 17.09 m/s

You've got a sign wrong somewhere. How can you coast uphill and end up faster at the top?

please tell me where i went wrong

(ho - h1) is not a positive number. You have taken it to be +4.9 m

To find the speed at which the cyclist crests the hill, we can use the principle of conservation of energy. At the bottom of the hill, the cyclist has both kinetic energy (due to the motion) and potential energy (due to the height). At the top of the hill, the kinetic energy and potential energy are interchanged but the total energy remains constant.

The total energy of the cyclist can be expressed as:

Initial energy = Final energy

The initial energy consists of kinetic energy (KE) and potential energy (PE) at the bottom of the hill:

Initial energy = KE + PE

The final energy consists of kinetic energy and potential energy at the top of the hill. Since the cyclist is coasting without pedaling, the final kinetic energy is zero:

Final energy = 0 + PE

To find the final velocity (Vf) at the top of the hill, we need to equate the two energies:

Initial energy = Final energy

KE + PE = 0 + PE

Since the cyclist is coasting without pedaling, the initial kinetic energy (KE) is given by 1/2 * m * V0^2, where m is the mass of the cyclist and V0 is the initial velocity:

1/2 * m * V0^2 + m * g * ho = 0 + 0

Here, ho is the initial height (4.9 m) and g is the acceleration due to gravity (9.81 m/s^2).

Simplifying the equation:

1/2 * V0^2 + g * ho = 0

Now, we can solve for Vf by rearranging the equation:

1/2 * V0^2 = - g * ho

V0^2 = - 2g * ho

V0 = √(-2g * ho)

Plugging in the values:

V0 = √(-2 * 9.81 * 4.9)

V0 = √(96.138)

V0 ≈ 9.805 m/s

Therefore, the speed at which the cyclist crests the hill is approximately 9.805 m/s.

Please note that there seems to be some confusion in the initial question as it states that the cyclist approaches the bottom of the hill at a speed of 14 m/s, but based on the calculations, the velocity is found to be approximately 9.805 m/s.