Differential equation of motion

[tomwilliam]

Homework Statement

I have to set up a differential equation of motion for a particle undergoing projectile motion, for which I know the initial position and velocity vectors

Homework Equations

The kinematics equations.

The Attempt at a Solution

Well I can easily take a(t) = -gj, integrate it wrt time to get v(t) = -gtj + C, then set time to zero, input my initial velocity vector and get v(t) = -gtj + v_0. Then I do the same thing to get position, integrating wrt time and inserting the original position vector to get:
r(t) = -1/2 gt^2 j + v_0t + r_0
I'm fairly sure this is enough to completely describe the projectile motion over the time interval...but I don't know exactly what constitutes a differential equation of motion. Can anyone help?
Thanks

 

[Andrew Mason]

Homework Statement

I have to set up a differential equation of motion for a particle undergoing projectile motion, for which I know the initial position and velocity vectors

Homework Equations

The kinematics equations.

The Attempt at a Solution

Well I can easily take a(t) = -gj

That is all you need. This can be written in the form of a differential equation where a is the second derivative of the displacement vector:

$$\frac{d^2\vec s}{dt^2} = -g\hat j$$

The solution to that equation describes the position of the projectile at time t. I am ignoring any changes in g due to altitude. If you want to take that into account what would you use instead of g?

AM

 

[tomwilliam]

Thanks.
On your question...if you wanted to take into account changes in g with altitude...I would think something like dg/dy...which could represent the change of the value of g with altitude, but then the equation wouldn't balance dimensionally...so I guess I don't know.

 

[Andrew Mason]

Thanks.
On your question...if you wanted to take into account changes in g with altitude...I would think something like dg/dy...which could represent the change of the value of g with altitude, but then the equation wouldn't balance dimensionally...so I guess I don't know.

Just apply Newton's law of gravitation:

$$F = -\frac{GMm}{r^2}\hat r$$

The acceleration is just $a = F/m = -\frac{GM}{r^2}\hat r$

Note, the magnitude of the force changes with r and the direction of the force changes with horizontal displacement.

AM