Final answer:
To clear Olympus Mons by firing a projectile horizontally from its summit, one would need an initial velocity of approximately 2659 m/s, calculated using the projectile's time in the air and the horizontal distance to clear based on Mars' gravitational acceleration.
Step-by-step explanation:
To determine the initial velocity required to clear Olympus Mons on Mars when firing a projectile horizontally from its summit, we need to apply principles of projectile motion. Given that the height of Olympus Mons is 25 km (25,000 meters) and we need to clear at least this distance to land on the surface; we can use kinematic equations to make this calculation. The force of gravity on Mars is approximately 3.71 m/s2, which is less than on Earth.
To calculate the time t it takes for the projectile to fall 25,000 meters, we use the kinematic equation:
h = 0.5 * g * t2
t = sqrt((2*h)/g)
Calculating this we find:
t = sqrt((2*25000m) / 3.71 m/s2)
t ≈ 116.2 seconds
Since we are firing the projectile horizontally, the initial vertical velocity is 0. The next step is to calculate the horizontal distance that needs to be cleared. If we consider the summit as the point of projection, the radius of the volcano (309 km or 309,000 meters) gives us the horizontal distance that must be cleared.
The horizontal initial velocity vi can then be calculated using:
horizontal distance = vi * t
vi = horizontal distance / t
Therefore:
vi = 309,000m / 116.2s
vi ≈ 2659 m/s
The initial velocity required to clear Olympus Mons is approximately 2659 m/s.