An interplanetary spaceship lands on a planet with a negligible atmosphere and fires three projectiles carrying exploratory bots at the planet, but at different initial velocities. After the bots land on the surface, their distances are measured and the data is recorded, as follows:
Velocity in m/s | Distance in m |
400 | 38,098 |
600 | 85,692 |
800 | 152,220 |
? | 300,000 |
At what speed should the projectile carrying the fourth bot be fired in order for it to land 300 km from the spacecraft?
For this problem, we need to understand the trajectory of the projectile. Since the atmosphere on the planet is weak, the trajectory is almost equivalent to the ballistic curve without any air drag. The distance, d, traveled by an object fired from a point on the ground, neglecting the curvature of the planet's surface, is given approximately by the following equation:
Where v is the initial velocity of the object, τ is the angle at which the object was fired, and g is the gravitational...