A 1250 kg car traveling at 20 m/s suddenly runs out of gas while approaching the valley shown in the figure below. What will be the car's speed as it coasts into the gas station on the other side of the valley? (Assume that no energy is loss to friction.)

file://dd-s-1/profiles/1104853/Desktop/10-p-013.gif

To find the car's speed as it coasts into the gas station on the other side of the valley, we can use the principle of conservation of mechanical energy.

First, let's identify the key information given in the problem:
- The mass of the car is 1250 kg.
- The initial velocity of the car is 20 m/s.
- The energy loss to friction is assumed to be negligible.

Now, let's consider the conservation of mechanical energy:
The initial mechanical energy of the car is equal to the final mechanical energy of the car, neglecting any energy loss due to friction. The mechanical energy consists of kinetic energy (KE) and gravitational potential energy (PE).

The initial kinetic energy of the car is given by:
KE_initial = (1/2) * mass * velocity^2

Substituting the values:
KE_initial = (1/2) * 1250 kg * (20 m/s)^2 = 250,000 Joules

The final mechanical energy of the car will consist only of gravitational potential energy since it has stopped due to running out of gas at the top of the hill. The potential energy is given by:
PE = mass * gravity * height

From the figure, we can approximate the height of the hill to be 50 meters.

Substituting the values:
PE = 1250 kg * 9.8 m/s^2 * 50 m = 612,500 Joules

Since there is no energy loss to friction, the initial mechanical energy is equal to the final mechanical energy:
KE_initial = PE

250,000 Joules = 612,500 Joules

To find the final velocity, we can rearrange the equation for kinetic energy:
KE_final = (1/2) * mass * velocity_final^2

Rearranging:
velocity_final^2 = 2 * (KE_final / mass)

Plugging in the values for KE_final and mass:
velocity_final^2 = 2 * (612,500 Joules / 1250 kg)
velocity_final^2 = 980 m^2/s^2

Taking the square root of both sides:
velocity_final = sqrt(980) m/s

Calculating the answer:
velocity_final = 31.3 m/s

Therefore, the car's speed as it coasts into the gas station on the other side of the valley will be approximately 31.3 m/s.