Finding safe distances from projectile bombardment over a mountain

[discoverer02]

An enemy ship is on the western side of a mountain island. The enemy ship can maneuver to within 2500 m of the 1800-m-high mountain peak and can shoot projectiles with an initial speed of 250m/s. If the eastern shoreline is horizontally 300m from the peak, what are the distances from the eastern shore at which a ship can be safe from the bombardment of the enemy ship?

I need to find the two angles at which the height of the projectile is 1800m at 2500m from the boad, but I'm having trouble relating all the different equations for Range, max height, and the other kinematic equations for motion in two dimensions.

A suggestions would be greatly appreciated.

 

[discoverer02]

OK, here's my latest approach. I want the height of the projectile to be 1800m at the same time it's moved 2500m horizontally.

The time at which it moves 2500m horizontally is 10sec/cos@. I then plug this into y = at^2 + vt + y(initial).

So, 1800m = -490m/(cos@)^2 + 2500m(sin@)/cos@ Am I correct so far?

If so, I'm at a loss as to how I should deal with the trigonometry.

 

[arcnets]

Hi discoverer02,
you forgot the factor 1/2 in your equation, but you fixed that in the next step. So I think your analysis is OK.

I tried more simple approaches, but none worked. So I think this is the way to do it. Next, you could of course substitute sin a = sqrt (1 - (cos a)^2). This will give a biquadratic equation for cos a, which has indeed 2 solutions. It's not difficult, but lengthy & boring. Sorry, I see no other way.

 

[discoverer02]

Hi arcnets,

Thanks for taking the time to look at the problem and posting a suggestion.

I'm still a little stumped by the trigonometry. If I use the identity you suggest, do I plug sqrt(1-(cos A)^2) into the quadratic equation when solving for 1/(cos a)? If not, then I can't figure out how to get rid of it.

Thanks again for your help.

discoverer02

 

[arcnets]

Originally posted by discoverer02
1800m = -490m/(cos@)^2 + 2500m(sin@)/cos@

Next (omitting units and using C = cos@):
1800 = -490/C^2 + 2500 sqrt(1-C^2)/C
1800 C^2 = -490 + 2500 C sqrt(1-C^2)
1800 C^2 + 490 = 2500 C sqrt(1-C^2)
1800^2 C^4 + 490^2 + 2*1800*490*C^2 = 2500^2 C^2 (1 - C^2)
1800^2 C^4 + 490^2 + 2*1800*490*C^2 = 2500^2 C^2 - 2500^2 C^4
4300 C^4 + (2*1800*490 - 2500^2) C^2 + 490^2 = 0.

That's the biquadratic beast. Solve.

 

[discoverer02]

arcnets,

Thanks again.

I thought maybe there was an easier way than squaring both sides that I wasn't seeing, but I guess not.

I guess I've become use to homework problems resolving themselves into neat little equations. Not always the case though.

 

[arcnets]

Maybe there's a nice, easy, and symmetric solution. But I don't see one.
There's an error in my last line, Should read
9490000 C^4 + ...