Another Physics Problem

[Borodog]

Much harder. Requires calculus. Not for the faint of heart.

A projectile is fired from the origin at an initial speed v_0 at an angle theta above the horizontal. If the force due to air drag on the projectile is of the form f = -bv, where b is the drag coefficient of the projectile, what is the trajectory of the projectile y(x)? (y is the vertical coordinate, x the horizontal.)

A camera at the origin is to stay pointed toward the projectile throughout its flight and focused on it. At what angle above the horizontal must the camera be pointed (as a function of time)? What is the distance to the projectile as a function of time?

Bonus credit: Include a graph of the trajectory for suitable values, as well as the angle and distance vs. time graphs.

BTW, this problem can be done completely analytically.

[holmansf]

Assuming the projectile stays low enough that we can approximate the acceleration due to gravity by a constant acceleration, the position function of the projectile satisifes

p''(t) = g - b p'(t)

where g is the acceleration due to gravity.

The general solution to this ODE is:

p(t) = tg/b -c e^{-bt} /b + a.

The initial conditions are p(0) = 0 and p'(0) = (v_0 cos(theta), v_0 sin(theta)). Plugging these in we get

0 = -c/b + a, g/b + c = (v_0 cos(theta), v_0 sin(theta))

implies

c = ba, g/b + ba = (v_0 cos(theta), v_0 sin(theta))

a =(b (v_0 cos(theta), v_0 sin(theta)) - g)/b^2.

So

p(t) = gt/b + (b (v_0 cos(theta), v_0 sin(theta)) - g)/b^2 (1 - e^{-bt})

Setting g = (0,-g_0), you can find all of the required quantities with a little algebra.

[Borodog]

p''(t) = g - (b/m)p'(t)

[thylacine]

[ QUOTE ]
p''(t) = g - b p'(t)

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[ QUOTE ]
p''(t) = g - (b/m)p'(t)

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No harm done: the advantage of not plugging in numbers.

[recipro]

I am just curious, but is it normal to use p(t) rather than r(t) for position? I've always used r(t)...

[SitNHit]

[ QUOTE ]
I am just curious, but is it normal to use p(t) rather than r(t) for position? I've always used r(t)...

[/ QUOTE ]

Im so happy i didnt take this class.

[holmansf]

ah well, doesn't matter much. Replace b by b/m in the solution.

[Borodog]

[ QUOTE ]
I am just curious, but is it normal to use p(t) rather than r(t) for position? I've always used r(t)...

[/ QUOTE ]

r(t) is the more common of the two; p(t) would make me think momentum. x(t) is what I would normally use, however.

[Borodog]

[ QUOTE ]
ah well, doesn't matter much. Replace b by b/m in the solution.

[/ QUOTE ]

True, but you still haven't answered any of the questions. /images/graemlins/wink.gif

[holmansf]

[ QUOTE ]
True, but you still haven't answered any of the questions.

[/ QUOTE ]

The rest is an easy exercise left to the reader. /images/graemlins/wink.gif Right back at ya.

[Borodog]

[ QUOTE ]
[ QUOTE ]
True, but you still haven't answered any of the questions.

[/ QUOTE ]

The rest is an easy exercise left to the reader. /images/graemlins/wink.gif Right back at ya.

[/ QUOTE ]

Try writing that on the test!

[gumpzilla]

[ QUOTE ]
[ QUOTE ]
[ QUOTE ]
True, but you still haven't answered any of the questions.

[/ QUOTE ]

The rest is an easy exercise left to the reader. /images/graemlins/wink.gif Right back at ya.

[/ QUOTE ]

Try writing that on the test!

[/ QUOTE ]

I handed in a knot theory assignment once where I got stuck proving an auxiliary lemma I thought I needed and gave 'er the old "I've discovered a truly marvelous proof, but this margin is too small to contain it." and carried on. It came back with a note scrawled in the margin saying "This counterexample fits."

[Borodog]

lol. That's awesome.

[thylacine]

[ QUOTE ]
[ QUOTE ]
[ QUOTE ]
True, but you still haven't answered any of the questions.

[/ QUOTE ]

The rest is an easy exercise left to the reader. /images/graemlins/wink.gif Right back at ya.

[/ QUOTE ]

Try writing that on the test!

[/ QUOTE ]

This is a poker forum. /images/graemlins/grin.gif

[holmansf]

[ QUOTE ]
[ QUOTE ]
[ QUOTE ]
True, but you still haven't answered any of the questions.

[/ QUOTE ]

The rest is an easy exercise left to the reader. /images/graemlins/wink.gif Right back at ya.

[/ QUOTE ]

Try writing that on the test!

[/ QUOTE ]

I was recently grading an analysis assignment and someone wrote that.