An interplanetary spaceship lands on a planet with negligible atmosphere and fires three projectiles at the same angle carrying exploratory bots, but at different initial velocities. After the bots land on the surface their distances are measured and the data recorded as follows:

At what speed should the projectile carrying the 4th bot be fired in order for it to land 300 km from the spacecraft?

Analysis:

For this problem we need to understand the trajectory of the projectile. Since the atmosphere on the explored planet is weak, the trajectory is almost equivalent to the ballistic curve without the air drag. The distance d traveled by an object fired from a point on the ground is approximately (neglecting the curving of the planet surface) given by the equation:

Where v is the initial velocity of the object, τ is an angle at which the object was fired and g is the gravitational force exerted by the planet on the object. Note that the angle τ and the gravitational force g do not change. Therefore define a constant . Then the distance on the explored planet can be explained in terms of the velocity by the equation:

Although d and v are not in the linear relationship, d and the square of v are. Therefore we can still apply the linear regression to determine the relationship between d and v.

Analysis using R:

Input:

source_code/6/speed_distance.r
trajectories = data.frame(
    squared_speed = c(160000,360000,640000),
    distance = c(38098, 85692, 152220)
)
model = lm(squared_speed ~ distance, data = trajectories)
print(model)

Output:

$ Rscript speed_distance.r 
Call:
lm(formula = squared_speed ~ distance, data = trajectories)
Coefficients:
(Intercept)     distance  
   -317.708        4.206

Therefore the relationship between the squared velocity and the distance is predicted by the regression to be:

The presence of the intercept term may be caused by the errors in the measurements or by other forces playing in the equation. Since it is relatively small, the final velocity should be estimated reasonably well. Putting the distance of 300km into the equation we get:

Therefore for the projectile to reach the 300km from the source, we need to fire it at the speed of 1123.157 m/s approximately.