At t = 0 on the planet Gravitus Increasicus, a projectile is fired withspeed v0 at an angle θ above the horizontal. This planet is a strangeone, in that the acceleration due to gravity increases linearly with time,starting with a value of zero when the projectile is fired. In other words,g(t) = βt, where β is a given constant. What horizontal distance doesthe projectile travel? What should θ be to maximize this distance?

The problem involves a projectile motion with a variable acceleration due to gravity. The horizontal and vertical motions can be analyzed separately.

1. Horizontal motion: The horizontal component of the velocity remains constant throughout the motion because there is no horizontal acceleration. The horizontal distance (d) covered by the projectile can be given by the formula:

   d = v0 * cos(θ) * t

   where v0 is the initial velocity, θ is the angle of projection, and t is the time of flight.

2. Vertical motion: The vertical component of the velocity changes with time due to the variable acceleration. The time of flight can be obtained by integrating the equation of motion from t=0 to t=T (time of flight):

   ∫from 0 to T (v0 * sin(θ) - 0.5 * β * t^2) dt = 0

   Solving this equation will give the time of flight in terms of v0, θ, and β.

3. Substituting the time of flight from step 2 into the equation from step 1 will give the horizontal distance covered by the projectile in terms of v0, θ, and β.

4. To maximize the horizontal distance, we need to differentiate the equation from step 3 with respect to θ and set the derivative equal to zero. Solving this equation will give the value of θ that maximizes the horizontal distance.

Please note that the actual calculations will depend on the specific values of v0 and β.