Projectile motion of a particle

[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?

 

[Doc Al]

Looks good to me.

 

[Nylex]

Cheers Doc! :)

 

[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 :(.

 

[Doc Al]

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 :(.

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).

 

[Nylex]

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).

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.