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

 

[rwisz]

A differential equation of motion is a mathematical equation that describes the relationship between the position, velocity, and acceleration of a particle over time. In this case, the kinematic equations that you have used to describe the projectile motion can be written in the form of a differential equation. The equation would be:

d^2r/dt^2 = -gj

This represents the second derivative of the position vector with respect to time, which is equal to the acceleration of the particle in the y-direction, as given by the acceleration due to gravity (-gj). This equation, along with appropriate initial conditions, can be solved to obtain the position and velocity of the particle at any given time during its motion.

In general, a differential equation of motion can be written as:

d^2x/dt^2 = F(x,t)

where x represents the position of the particle, t represents time, and F(x,t) represents the net force acting on the particle. This equation can be used to describe the motion of particles in various systems, such as projectile motion, oscillatory motion, and circular motion.

In summary, the differential equation of motion for a particle undergoing projectile motion is d^2r/dt^2 = -gj, where r is the position vector and t is time. This equation, along with initial conditions, can be used to fully describe the motion of the particle.