Firing mortar and cliff edge, Feynman Lectures 4.17

[Jabedi13]

Homework Statement

"A mortar emplacement is set 27,000 ft horizontally from edge of a cliff that drops 350 ft down from level of mortar...It is desired to shell objects concealed on the ground behind the cliff. What is the smallest horizontal distance d from the cliff face that shells can reach if fired at a muzzle speed 1000 ft/s"
Problem 4.17 from Exercises for the Feynman Lectures on Physics

Homework Equations

x = v*cos(theta)*t
v(y) = v*sin(theta) - g*t
y = v*sin(theta)*t -(1/2)*32*t^2

The Attempt at a Solution

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This may sound stupid, but I am really stumped on this one. If my basic understanding of the concept is correct, I should be trying to use 27,000 - d = x = v*cos(theta)*t Other attempts gave me incorrect answers.
With this equation I have three unknowns: d, theta, and time. I know I am probably missing something simple and stupid but I can not figure out how to use the projectile motion equations to eliminate the other variables.

 

[phyzguy]

First off, it should be 27000 + d, not 27000 - d, because d is past the edge of the cliff. You know y, because you know the final height. So you can use the y equation to write t in terms of theta and d. Then you can plug this into the x equation to eliminate t, giving you one equation relating theta and d. Then you use calculus to find the value of theta that minimizes d. Why don't you try this and post your attempts.

 

[TSny]

Hello, Jabedi13. Welcome to PF!

Draw a sketch of the setup and try drawing various trajectories. Is there anything special about the trajectory that gives the minimum distance between the cliff face and the point of landing?

 

[Jabedi13]

I have tried it a couple times, solving y for t. I end up with a complicated quadratic formula. Using that in the x equation gives a messy derivative that I probably got wrong and am not sure how to set to zero. Am I missing something?

Tried uploading pictures, didn't work. I will try to upload and scan them when I get home.

 

[TSny]

I have tried it a couple times, solving y for t. I end up with a complicated quadratic formula. Using that in the x equation gives a messy derivative that I probably got wrong and am not sure how to set to zero. Am I missing something?

You don't need to use any calculus.

Hint: By sketching various trajectories, can you notice anything regarding how far the shell should miss the top edge of the cliff in order to minimize the horizontal position of impact at the ground level?