While traveling to school at 27 m/s your car runs out of gas 15 hm from the nearest gas station. If the station is 16m above your current elevation, how fast will the car be going when it reaches the gas station? Ignore friction.

To determine the speed of the car when it reaches the gas station, we can use the principle of conservation of energy. The total mechanical energy of the car remains constant throughout its motion, neglecting any friction or external forces.

First, let's convert the given distance to meters. 15 hm is equivalent to 1500 meters (since 1 hectometer = 100 meters).

Next, we need to find the potential energy gained by the car as it goes up the 16-meter elevation difference. The potential energy can be calculated using the formula:

Potential energy = mass × gravitational acceleration × height

Since only the elevation changes, we can ignore the car's mass. The gravitational acceleration is 9.8 m/s². Plugging in the values, we have:

Potential energy = 0 (mass) × 9.8 m/s² × 16 m = 0 Joules

Now, let's consider the initial kinetic energy (KE) of the car:

Initial kinetic energy = 0.5 × mass × velocity^2

Again, we can ignore the car's mass for this calculation. The given initial velocity is 27 m/s. Plugging in the values, we have:

Initial kinetic energy = 0.5 (mass) × 27 m/s)^2
= 0.5 × (27 m/s)^2
= 0.5 × 729 m^2/s^2
= 364.5 Joules

Since the total mechanical energy remains constant, the initial kinetic energy must equal the potential energy gained during the ascent. Therefore:

Initial kinetic energy = Potential energy

364.5 Joules = 0 Joules

Now, let's find the velocity of the car when it reaches the gas station. At this point, the potential energy will be zero, and only the kinetic energy will remain.

Final kinetic energy = 0.5 × mass × final velocity^2

Plugging in the values, we have:

0.5 × (final velocity)^2 = 364.5 Joules

Simplifying the equation, we get:

(final velocity)^2 = 364.5 Joules / 0.5
= 729 m^2/s^2

Taking the square root of both sides, we find the final velocity:

final velocity = √(729 m^2/s^2)
= √(729) m/s
= 27 m/s

Therefore, the car will be traveling at a speed of 27 m/s when it reaches the gas station.