Rate of change of acceleration question?

Question

OK so it's easy enough to deal with an object being under constant acceleration, but how do we deal with an the final speed of an object after an amount of time when the acceleration is increasing . So let's say we have a ball in space of 1 kilogram, and it's being drawn towards a mass of 500000 kg, and its distance is 150 km, and we say that the larger mass doesn't move towards the smaller one. How long would it take the masses to collide and what would the final velocity of the smaller mass be ? (i.e. the velocity of the small mass just before it hit the larger one)

Any clues ?

Thanks.

Answer 1

I'm not going to work it out, but you need to set up a differential equation.  The acceleration (y") is a function of the displacement (y)

Since Gravitational force = GMm/r^2

y" = a = F/m = -G(M)/y^2

Answer 2

You would need to come up with an equation that describes the acceleration over time.  I can't quite remember gravitational acceleration's equation, but it's something like a=(mM)/(r^2)  where m and M are the two masses.

From this you would need to integrate with respect to time to get the equation for velocity.  Find what time is when r=0 with the acceleration equation and that is your collision time. Plug that into velocity and it will give you your final V.

Answer 3

The velocity is found by energy methods:

GPE = GMm(1/R - 1/D) where R is the radius of the mass and D = 150 km

Equating this to the kinetic energy of the ball when it strikes the surface (½mV²) gives

V = √(2GM(1/R - 1/D)

For the time you'll have to do some integration........

Answer 4

If you know the three laws of Kepler and have suufficient IQ, then you can figure exact answer yourself, Allah willing. The answer to this question was known BEFORE Newton invented calculus, let alone differential equations.

At first try to figure out velocity and period of circular motion about point mass M at distance r, use formula for centripetal accelartion and universal law of gravity:

a = 1/2 v²/r = GMm/R²

Then try to see what happens if initial velocity is less than required for circular motion, new orbit is ellipse, right?

No try to see what happens to elliptic orbit as initial velocity tends to zero. What happens to period? What happens to sectoral area swept?

If you are interested in learning wonderful ways of Allah, try do all this at least partilly yourself, and add additional details, then I will see to it that you get to the proper solution, which is maginificent.

Or better yet study the holy Koran.

Answer 5

You are fortunate to have Mullah Abdullah deem your question worthy of his attention.  He has learned more from the study of the holy Koran than most mortals have learned from the study of calculus.

Mullah Abdullah methods will give you the exact solution for both time and velocity, and you will learn much should you choose to learn from his example.

Ponder his advice.  If you are still unable, and should Mullah Abdullah becomes too preoccupied in his important work.  I will post the correct solution.