We can use the laws of physics to solve this problem. The key here is to recognize that the initial kinetic energy of the car (due to its speed) is converted into potential energy as it goes up the ramp, and then back into kinetic energy as it falls back down.
Using conservation of energy, we can set the initial kinetic energy equal to the final potential energy at the maximum height:
(1/2) * m * v^2 = m * g * h
where m is the mass of the car, v is its initial speed (30 m/s), g is the acceleration due to gravity (9.81 m/s^2), and h is the maximum height.
Simplifying the equation and solving for h, we get:
h = (v^2 * sin^2(theta)) / (2 * g)
where theta is the angle of the ramp (37 degrees in this case).
Plugging in the known values, we get:
h = (30^2 * sin^2(37)) / (2 * 9.81) ≈ 79.1 meters
So the car will reach a height of approximately 79.1 meters off the ground.
To find the maximum height of the gorge that the car could clear, we need to look at the horizontal distance that the car travels while in the air. This distance is given by:
d = v * t
where t is the time that the car spends in the air. We can find t by setting the initial potential energy (at the maximum height) equal to the final kinetic energy (at the end of the jump):
m * g * h = (1/2) * m * (v_f^2)
where v_f is the final velocity of the car at the end of the jump (when it lands back on the ground). We can solve for v_f using:
v_f = sqrt(2gh)
where h is the maximum height (found earlier). Plugging this into the conservation of energy equation, we get:
t = sqrt(2h/g)
Now we can plug in the known values for v and theta to get:
d = 30 * sqrt(2h/g) * cos(theta)
Plugging in the values for h and theta, we get:
d = 30 * sqrt(2*79.1/9.81) * cos(37) ≈ 187.2 meters
So the maximum height of the gorge that the car could clear is approximately 187.2 meters.