Just as a car tops a 51 meter high hill with a speed of 84 km/h it runs out of gas and coasts from there, without friction or drag. How high, to the nearest meter, will the car coast up the next hill?

v =84 km/h= 23.33 m/s.

mv^2/2+mgh = mgH.
H = h +v^2/2g =51 +27.78 =78.78 m.

To determine how high the car will coast up the next hill, we can use the principle of conservation of mechanical energy. At the top of the first hill, the car has potential energy due to its height and kinetic energy due to its speed. As it coasts up the next hill, its kinetic energy will convert to potential energy until it reaches a maximum height.

Here's how we can solve the problem:

1. Convert the speed from km/h to m/s:
Speed = 84 km/h = (84 * 1000 m) / (3600 s) = 23.3 m/s

2. Calculate the initial potential energy at the top of the first hill using the car's mass and height:
Potential Energy at the top of the first hill = mass * gravity * height
Let's assume the mass of the car is 1000 kg and gravity is approximately 9.8 m/s^2:
Potential Energy = 1000 kg * 9.8 m/s^2 * 51 m = 499,800 J (Joules)

3. Calculate the final potential energy at the maximum height the car reaches:
Since there is no friction or drag, the total mechanical energy (kinetic energy + potential energy) is conserved. Therefore, the final potential energy will be equal to the initial potential energy:
Final Potential Energy = 499,800 J

4. Calculate the height of the next hill using the final potential energy:
Final Potential Energy = mass * gravity * height
Rearranging the equation to solve for height:
height = Final Potential Energy / (mass * gravity)
height = 499,800 J / (1000 kg * 9.8 m/s^2) ≈ 51 meters

Therefore, the car will coast up the next hill to a height of approximately 51 meters, the same as the height of the first hill.