# Projectile motion of a particle

1. Nov 21, 2004

### Nylex

Some particle is given an initial velocity, u at an angle θ to the horizontal. I'm asked to find (as a function of θ): 1. the range of the particle, X, 2. the maximum altitude reached, Y and the time taken to reach maximum altitude, T.

First I resolved u into components: uy = usin θ, ux = ucos θ.

For 1, I said x = ucos θ.t (since the horizontal motion is unaccelerated, no air resistance).

To work out t, I used s = ut + (1/2)at^2 for the vertical motion, setting s = 0. For t != 0, I got t = (2usin θ)/g

=> X = (ucos θ.2usin θ)/g = (2u^2.sin θcos θ)/g = (u^2.sin 2θ)/g

For 2, I used the fact that v = 0 when the particle reaches its maximum height and the equation v^2 = u^2 + 2as.

=> Y = (1/2g)(uy)^2 = (1/2g)(usin θ)^2

For 3, I again used v = 0 at maximum height, but used v = u + at

=> T = (usin θ)/g

Then, I'm asked to work out the projectile's velocity as a function of time.

v(t) = [(vx)^2 + (vy)^2]^1/2

vx = ucos θ
Using v = u + at, vy = usin θ - gt

v(t) = [(ucos θ)^2 + (usin θ - gt)^2]^1/2

v(t) = [(ucos θ)^2 + ((usin θ)^2 - 2gtusin θ + (gt)^2]^1/2

v(t) = [u^2.(cos^2 θ + sin^2 θ) - 2gtusin θ + (gt)^2]^1/2

v(t) = [u^2 - 2gtusin θ + (gt)^2]^1/2

Is this correct?

2. Nov 21, 2004

### Staff: Mentor

Looks good to me.

3. Nov 21, 2004

### Nylex

Cheers Doc! :)

4. Nov 23, 2004

### Nylex

Hmm, I have to plot v(t) vs. t, for a given u and θ and all I get is a straight line. It doesn't seem right to me for some reason :(.

5. Nov 23, 2004

### Staff: Mentor

Well, that can't be right. (I assume you are plotting the magnitude of the velocity.) Try it again! It starts out with its maximum value of u ... decreases to a minimum value of $u cos\theta$ (at the top of the motion)... then increases again back to the original value (when it's back to the starting height).

6. Nov 23, 2004

### Nylex

Thanks again Doc. Yes, I'm plotting the magnitude of the velocity. My working for v is in my first post and I'm not sure what's wrong with it. Maybe I shouldn't have factorised u^2.cos^2 θ + u^2.sin^2 θ by u^2, but it shouldn't make a difference.