Need help - projectile motion with air resistance

[don_anon25]

Need urgent help -- projectile motion with air resistance

Consider a projectile fired vertically in a constant gravitational field. For the same inital velocities, compare the times required for the projectile to reach its maximum height (a) for zero resisting force and (b) for a resisting force proportional to the instantaneous velocity of the projectile.

For a, I get t=-v0/a.
For b, I have the equation F= m dv/dt=kmv-mg, which leads me to
dv/dt=kv-g.
Is this correct...If so, I can take it from here.
If not what should it be?? What should my initial condition be to solve for the constant? Should it be v(t=0)=v0?

Thanks much!

 

[Fermat]

The resisting force is simply kv, rather than mkv and it is acting in the same direction as the gravitational force, mg.

Initial/end conditions are v_i = v_o and v_f = 0.

 

[quicknote]

Hello,

It seems like you are on the right track with your equations. For the case with zero resisting force, your formula for the time to reach maximum height is correct. For the case with air resistance, your equation for the velocity is also correct. To solve for the constant k, you can use your initial condition of v(t=0)=v0. This means that at the initial time, the velocity of the projectile is equal to its initial velocity. You can then solve for k using this condition and your equation for velocity.

Once you have the value of k, you can use it to solve for the time to reach maximum height in the same way as you did for the case with zero resistance. However, it is important to note that the presence of air resistance will cause the projectile to reach a lower maximum height and take longer to reach it compared to the case with no resistance.

I hope this helps and good luck with your calculations.