Starship + 2 different planets?

[good.you?]

Starship + 2 different planets?

Homework Statement

The problem from my textbook is:
Inside a starship at rest on earth, a ball rolls off the top of a horizontal table and lands a distance D from the foot of the table. This starship now lands on the unexplored Planet X. The commander, Captain Curious, rolls the same ball off the same table with the same initial speed as on Earth and finds that it lands a distance 2.76D from the foot of the table. What is the acceleration due to gravity on Planet X?

Homework Equations

The 4 basic equations of motion:
V=Vi + a x delta t
S=Si + Vi x delta t + 1/2at^s
S=Si + 1/2(V+Vi)delta t
V^2= Vi^2 + 2a(S-Si)

The Attempt at a Solution

I know the initial velocity is the same in both cases, I figure you have to substitute the initial velocity in one with an equation with the acceleration on earth. I'm kind of lost from this point.
What I really need is some one to explain how to solve it.

 

[kuruman]

This is a projectile motion problem. You need to find an expression for the range of the projectile given that it has initial velocity v₀ only in the horizontal direction and starts at height h above ground. Can you find such an expression?

 

[good.you?]

I'd have to rearrange equations and substitute them into one another and I wouldn't know where to start.

 

[kuruman]

Start by writing expressions for the horizontal position of the ball at any time t and for the position of the ball above ground at any time t. Eliminate the time and find an expression giving the dependence of the range on the vertical height and the initial speed.

 

[rwisz]

I would approach this problem by first identifying the known and unknown variables. We know the distance D on Earth and 2.76D on Planet X, and we are trying to find the acceleration due to gravity on Planet X.

Next, I would use the equation S=Si + Vi x delta t + 1/2at^2 to solve for the acceleration on Earth. We know the initial velocity (Vi) is the same in both cases, and we can plug in the known distance D and the time it takes for the ball to land (which is the same on both Earth and Planet X). This will give us the acceleration due to gravity on Earth.

Once we have the acceleration on Earth, we can use the equation V^2= Vi^2 + 2a(S-Si) to solve for the acceleration on Planet X. We know the initial velocity and the distance (2.76D) on Planet X, and we can plug in the acceleration we found on Earth. This will give us the acceleration due to gravity on Planet X.

It's important to note that this solution assumes the same initial conditions and no external forces acting on the ball. If there are any other factors at play, the results may vary.