# Motion in Three-Space

• Mar 10th 2009, 01:20 AM
jffyx
Motion in Three-Space
A gun has a muzzle speed of 90 meters per second. What angle of elevation should be used to hit an object 150 meters away? Neglect air resistance and use http://hosted.webwork.rochester.edu/...325ddf14f1.png as the acceleration of gravity.

Thanks!
• Mar 10th 2009, 01:54 AM
mr fantastic
Quote:

Originally Posted by jffyx
A gun has a muzzle speed of 90 meters per second. What angle of elevation should be used to hit an object 150 meters away? Neglect air resistance and use http://hosted.webwork.rochester.edu/...325ddf14f1.png as the acceleration of gravity.

Thanks!

It will depend on what height you want to hit the object at. Or is it to be assumed that the object is at ground level ....?
• Mar 10th 2009, 02:46 AM
jffyx
Quote:

Originally Posted by mr fantastic
It will depend on what height you want to hit the object at. Or is it to be assumed that the object is at ground level ....?

Oh. It is to be assumed that the object is at ground level.
• Mar 10th 2009, 05:51 AM
mr fantastic
Quote:

Originally Posted by jffyx
Oh. It is to be assumed that the object is at ground level.

Substitute the data into equation (7) here: Projectile Motion

Now solve this equation for $\theta$.
• Mar 10th 2009, 09:00 AM
Soroban
Hello, jffyx!

Quote:

A gun has a muzzle speed of 90 meters per second.
What angle of elevation should be used to hit an object 150 meters away?

The equations for projectile motion are: . $\begin{array}{ccc}x \;=\; (v_o\cos\theta)t \\ y \;=\; h_o + (v_o\sin\theta)t - 4.9t^2 \end{array}$
. . where: . $\begin{Bmatrix}h_o &=& \text{initial height} \\ v_o &=& \text{initial speed} \\ \theta &=& \text{angle of elevation} \end{Bmatrix}$

We are given: . $h_o = 0,\;v_o=90$

The equations are: . $\begin{array}{cc}x \;=\;(90\cos\theta)t & {\color{blue}[1]} \\ y \;=\;(90\sin\theta)t - 4.9t^2 & {\color{blue}[2]} \end{array}$

Since $x = 150$, [1] becomes: . $(90\cos\theta)t \:=\:150 \quad\Rightarrow\quad t \:=\:\frac{5}{3\cos\theta}$

Substitute into $y = 0\!:\;\;(90\sin\theta)\left(\frac{5}{3\cos\theta}\ right) - 4.9\left(\frac{5}{3\cos\theta}\right)^2 \:=\:0$

which simplifies to: . $1350\tan\theta - 122.5\sec^2\!\theta \:=\:0$

. . $1350\tan\theta - 122.5(\tan^2\!\theta + 1) \:=\:0 \quad\Rightarrow\quad 122.5\tan^2\!\theta - 1350\tan\theta + 122.5 \:=\:0$

Quadratic Formula: . $\tan\theta \;=\;\frac{1350 \pm\sqrt{1,\!762,\!475}}{245} \;=\;\begin{Bmatrix}0.091500952 \\ 10.92890771 \end{Bmatrix}$

. . $\tan\theta \,=\,0.091500952\quad\Rightarrow\quad \theta \:\approx\:5.228^o$

• Mar 10th 2009, 10:06 AM
Mentia
Note how Soroban got two solutions. You could also fire the gun at about 84.77 degrees and hit the target.