Projectile practice

You have a cannon a distance $d$ before the start of a hill. The hill goes up at an angle $\theta$ from the ground. What angle and velocity must you shoot a cannon ball so that it lands a distance $s$ up the hill?(The distance s is measured from the bottom of the hill, i.e. the distance along the hill).

#Mechanics

Easy Math Editor

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`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
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Comments

I think there are infinite solutions to this problem. There are infinite parabolas passing through (0,0) and $(d+sCos\theta,sSin\theta)$ , the cannon at (0,0).

Niranjan Khanderia - 5 years, 10 months ago

Well, I believe that the gravitational field is fixed, so you may assume that acceleration is $a$ ....

Calvin Lin Staff - 5 years, 10 months ago

Niranjan sir is right since angle and velocity both are upto us there are infinite possibilities.One of them has to be given for a finite(2 if velocity is given[not always]) number of solutions.

Krishna Sharma - 5 years, 10 months ago

@Krishna Sharma – Oh yes that's true. For some reason, I though the angle of the cannon was fixed (when it's just the angle of the hill that is fixed).

So, what is the relationship between $\theta$ and $v$ ?

Calvin Lin Staff - 5 years, 10 months ago

I got $\large v= (d+s\cos\theta)\sqrt{\dfrac{g}{2(\sin 2\alpha)(d+s\cos\theta) - s\sin\theta\cos^2\alpha}}$ where $v$ and $\alpha$ is the required velocity and angle. As you say , there are infinitely many solutions. Is this correct?

This question is given to me by @Josh Silverman Sir.

Nihar Mahajan - 5 years, 10 months ago

The above equation has two unknowns. Only one equation. Thus infinity solutions.

Niranjan Khanderia - 5 years, 10 months ago

@Niranjan Khanderia – Yes , that was what I was saying too.

Nihar Mahajan - 5 years, 10 months ago

Hey guys, I've made this into a well-constrained problem, here

Josh Silverman Staff - 5 years, 10 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$