# Projectile motion with air resistance question?

As part of my course I've been asked to investigate the trajectory of a tennis ball when the ball is being 'served'.

Now that part isn't too tricky, it simply involves newtons equations of motion.

However, as an extension to this investigation we have been asked to incoorporate Air Resistance. Now I am aware that air resistance is a function of velocity and so I cant simply just add an equation of air resistance into newtons equations of motions and expect it to work.

I am sure that involving air resistance with projectile motion involves differential equations.

I know how to work with differential equations and I know how to work with newtons equations of motions, the only thing I dont know is how to incorporate both the equations of motions and the differential equations for air resistance. Is there anyone who can help me develop a model for the tennis balls trajectory where air resistance is to be considered

thanks alot!

### 1 Answer

- Anonymous9 years agoBest Answer
Start with Force = mass*acceleration in the form a = F/m. Remember that acceleration is the differential of velocity so dV/dt = F/m. You must work on horizontal and vertical motion separately.

The force involved horizontally is only air resistance so dV/dt = -k*(Vh)/m

The forces involved vertically are gravity and air resistance so

dV/dt = [-g + k*(Vv)]/m

(g negative and k positive if you are measuring upwards. Reverse signs if not.)

Edit. However, both will be negative if the ball is rising.