Force of gravity in terms of time, rather than distance.

[Scottmeister]

My friends and I have a problem that seems simple enough, but has proven to be pretty hard. We need some help.

Suppose you have an object in space with no initial velocity relative to the Earth.

Now, due to the Earth's gravity, the object will begin to accelerate towards the Earth instantly.

We need to find the velocity of the object after any given time; so, we need an equation for the velocity of the object in terms of time t and the original distance r.

Please note: Acceleration due to gravity does NOT equal -9.81m/s^2 for this problem. acceleration due to gravity is -GM/r^2 and r is decreasing as t increases because the object is accelerating towards the Earth at a faster and faster rate.

Now, we realize this will probably take some differential equations to solve, but we can't find what differential equations we need to solve.

Any help at all is appreciated.

Thank you!

 

[Nabeshin]

Basically identical question answered here:
https://www.physicsforums.com/showthread.php?t=364781

Cheers!

 

[Scottmeister]

Thank you!

 

[Scottmeister]

Basically identical question answered here:
https://www.physicsforums.com/showthread.php?t=364781

Cheers!

I read over that other thread. I couldn't quite understand though. Did you ever find equations for the acceleration of gravity with respect to time, or did you just approximate the values?

 

[pmacias]

I understand your problem and I am happy to provide some guidance. To solve this problem, we need to use Newton's Second Law of Motion, which states that the net force on an object is equal to its mass multiplied by its acceleration. In this case, the net force is the force of gravity, and the acceleration is the rate at which the object's velocity is changing over time. So, we can write the equation as F = ma, where F is the force of gravity, m is the mass of the object, and a is the acceleration.

To find the velocity of the object after a given time, we can use the equation v = u + at, where v is the final velocity, u is the initial velocity (which is zero in this case), a is the acceleration, and t is the time. However, as you mentioned, the acceleration due to gravity is not a constant in this case, as the distance between the object and the Earth is changing.

To account for this, we can use the formula for the force of gravity, F = -GMm/r^2, where G is the gravitational constant, M is the mass of the Earth, m is the mass of the object, and r is the distance between the object and the Earth. We can rearrange this equation to solve for the acceleration, a = -GM/r^2. Now, we can substitute this into our equation for velocity, v = u + at, to get v = u - (GM/r^2)t.

To solve for the velocity at any given time, we also need to know the initial distance, r. So, the final equation for the velocity of the object in terms of time and distance is v = -GM/r^2t. This equation takes into account the changing distance between the object and the Earth as it accelerates towards it.

As for the differential equations, we can use the equation F = ma and the formula for the force of gravity to create a differential equation that describes the motion of the object. This equation will involve the acceleration, velocity, and distance variables, and can be solved using calculus techniques.

I hope this helps you and your friends in solving your problem. Remember to always use the appropriate equations and units when solving scientific problems. Good luck!