Differential Equations: The Rocket Problem

It's come to my attention that the solution to the differential equations problem to do with rocket flight (given about 11 months ago) was never presented, so let me address that now. The problem, to recap, was:

Find the x- and y-coordinates of the points on the trajectory of a rocket shot at an angle of 80 degrees with an initial velocity of 100,000 f/s if the air resistance is 0.01mv. Find the value of x and y after 10 seconds.

We have:

Θ = 80 deg, k = 0.01mv, v = 10^5 m/s and t = 10s

The differential equation can be written:

m(d^2x/dt^2) = - k (dx/dt)/ v = - (0.01mv)x’/ v = - 0.01mx’

Similarly:

my” = -mg – (0.01mv)y’/v

where as usual y" denotes the 2nd derivative wrt time, and y' the 1st derivative.

Whence:

my" = = -mg – 0.01my’

Yielding the simultaneous pair
:
x” + 0.01x’ = 0

y” + 0.01 y’ = -g

We know:

x’_o = v_o cos (Θ) = v_o cos (80) = (10^5) cos 80

y’_o = v_o sin (Θ) = v_o sin (80) = (10^5) sin 80

General Solutions are:

x = c1 + c2 exp(-0.01t)

y = c3 + c4(exp( -0.01t) - 100 gt

x’ = -0.01c2 exp(-0.01t)

y’ = -0.01c4 exp(-0.01t) - 100 g


For initial conditions, at t = 0, x = 0 so:

0 = c1 + c2 exp(-0.01t)

But:

x’_o = v_o cos (80) = (10^5) cos 80 =
-0. 01c2 exp(-0.01t)

Then: c2 = - (10^7) cos 80

So: 0 = c1 + (- (10^7) cos 80) or c1 = (10^7) cos 80


At t = 0, y = 0:

Then: 0 = c3 + c4 exp(-0.01t)

And: y’_o = v_o sin (80) = (10^5) sin 80 =
-0.01c4exp(-0.01t) - 100 g

So: c4 = -(10^7) sin 80 – 10000g

therefore, c3 = - c4 exp(-0.01t) = 10^7 sin 80 + 10000g


Now, to obtain the x and y-coordinates:

First, the x-coordinate:

x = c1 + c2 exp(-0.01t) = 10^7 cos 80 –
(10^7 cos 80)exp( -0.01t)

x = 10^7 cos 80 [1 - exp(-0.01t)] =
10^7 (0.17365) [1 - exp(-0.01t)]


or: x = 1736500 [1 - exp(-0.01t)]


The y-coordinate:


y = c3 + c4 exp(-0.01t)

y = 10^7 sin 80 + 10000g – 10^7 sin 80
+ 10000g(exp(-0.01t))

y = 10^7 sin 80 + 10000g[1 - exp( -0.01t)]
– 100 gt

y = [10^7 (0.9848) + 320,000][1 - exp(-0.01t)] – 100 gt

y = 1016800(1 - exp(-0.01t) – 100 gt


After 10 seconds (t = 10):


x = 1736500 [1 - exp(-0.01t)] = x = 1736500 [1 - exp (-0.01(10))]

x = 165, 200 ft. (= 31. 3 miles)

y = 1016800[1 - exp (-0.01(10)] – 100(32)(10)

y = 930,580 ft. (= 177 miles)