Projectile Motion and Human Cannonball

[micnike1]

Homework Statement

Hi,
I am doing an end of the year calculus presentation on projectile motion in human cannonballs. As an introduction to the projectile motion equations, my group found this problem:

Blammo is to be fired from 5mabove ground level with a muzzle velocity of 35m/s over a flaming wall that is 20 m high and past a 5-m-high shark pool. To make the feat impressive, the pool will be made as long as possible. Your job as Blammo’s manager is to determine the length of the pool, how far to place the cannon from the wall, and what elevation angle to use to ensure that Blammo clears the pool.

The original problem with a figure of it can be found http://higheredbcs.wiley.com/legacy/college/anton/0471482730/calc_horizons/blammo.pdf"

I'm not a physics student, so thank you for any help!​

Homework Equations

I'm sorry, but I couldn't figure out the latex equations...Hope these are clear enough.

Projectile Motion Formulas
R=(v_o²/g)sin(2q)
r(t)=v_0xti+(v_oyt-.5gt²j

which, if I'm correct (??) breaks down into:
x(t)=x_o+v_oxt
y(t)=y_o+v_oyt-.5gt²​

(Are there equations I'm missing for this problem that would be more useful/better than these??)
​

The Attempt at a Solution

Okay...Here we go:

R=(v_o²/g)sin(2q)

For the range of the cannon (R) to be maximum, the angle of elevation (q) must equal 45^o.
R=(35²/9.798)sin(2*45)
R=125.026m​

x(t)=x_o+v_oxt
y(t)=y_o+v_oyt-.5gt²

@45^o, v_ox=v_oy.
v_ox=v_ocos(q)
v_ox=v_oy=24.749m/s

I placed the wall of fire at x=0. So, solving for x_o would give the distance from the wall the cannon must be placed. (Correct??)
0=x_o+v_oxt
t=-x_o/v_ox
Sub t into the y(t) equation:
y(t)=20=5+v_oyt-.5gt²
15=-x_o-4.899(-x_o/24.749)²
x_o=-17.43m
​

R+x_o=length of pool

The pool is 107.596m long.​

So, I think I have solved this correctly. One thing I'm worried about is the assumption of q=45^o. I know this makes R the greatest, but does it make the length of the pool the greatest? If the angle was steeper, could the cannon be placed closer to the fire, making the pool be a greater part of the path?

Any help or confirmation of my answers would be greatly appreciated. Thank you so much!

 

[nvn]

micnike1: Your equations and math are correct (for q = 45 deg), except g appears to be inaccurate. However, your equations are not general, because they assume voy = vox, which is not true in the general case. Furthermore, your worry is well founded, because your answer is incorrect. After making your equations general (as mentioned above), how about if you check your assumption for q? Try plugging in a larger value for q, but less than 53 deg, and see what happens.

 

[micnike1]

Thanks nvn.

I've spent about an hour now trying to manipulate the equations into a general form. But I always end up with two variables (q and x_o) and am not able to substitute one for the other. Am I missing something here? Maybe I've misunderstood what you mean...

 

[nvn]

Your variables sound fine, if you did not use voy = vox in your new derivation. Did you? In post 1, it appears you said voy = vox.

 

[micnike1]

You're right, I did use v_oy=v_ox to cancel things out in the step 15=voy(-xo/vox)-.5g(-xo/vox)^2. Not using that assumption, I get down to 15=-x_otan(q)-.5g(-x_o/(v_ocos(q)))².

Where do I go from here. I'm like completely lost in this problem now...
Thanks again,
Mike

 

[nvn]

Excellent, micnike1. Also rederive your first equation under relevant equations in post 1. Then proceed as you did in post 1 to solve the problem. Also see post 2.