dynamics??

Question

A 5000 kg interceptor rocket is launched at an angle of 44.7degrees. The thrust of the rocket motor is 140,700 N. A)find an equation y(x) that describes the rockets's trajectory. b) what is the shape of the trajectory? c)At what elevation does the rocket reach the speed of sound, 330 m/s.

Answer 1

We are given enough information to calculate the equation for acceleration. From there we can find the other equations using calculus.

In order to find the other equations we need to make a few assumptions.
1. neglect friction of air
2. assume initial velocity is 0
3. assume initial altitude is 0
4. assume no wind

Now that we have this we figure out the acceleration equation.
We have two equations, one for x and one for y.

F = ma will be important.
a = F/m

We can make use of these using trigonometry. I will let you do the calculator work.

===Find a_x
T= thrust
F_x = T * cos(deg)
F_x = 140700 * cos(44.7)

a_x = F_x / m

a_x = 140700 * cos(44.7) / 5000

===Find a+y
T = thrust
g = accel. gravity

F_y = T*sin(deg) - mg
F_y = 140700 * cos(44.7) - 5000*9.81

a_y = F_y/m

a_y = (140700 * cos(44.7) - 5000*9.81)/5000

==Find V

to find velocity, we integrate.

We know that a is the derivative of V so we integrate with respect to time.


a_x = dV_x/dt =140700 * cos(44.7) / 5000

V_x(t) = int(140700*cos(44.7)/5000 dt)

V_x(t) = (1407/50) cos(44.7) * t



a_y = dV_y/dt = (1407 * cos(44.7) - 50*9.81)/50

V_y(t) = int[1407 * cos(44.7) - 50*9.81)/50 dt]

V_y(t) = [1407 * cos(44.7) - 50*9.81)/50] * t


=== Finding y(x)
To do this we first must find x(t). This gives us x as a function of t.
We are able to get y(t) from our equations. So, we use our x(t) to solve for 1 and plug that back into y(t) to get y(x).

V_x(t) = dx/dt = (1407/50) cos(44.7) * t
integrate just like before

x(t) = int((1407/50) cos(44.7) * t dt)

x(t) = (1/2)(1407/50) cos(44.7)*t^2
x(t) = 14.07*cos(44.7)*t^2

V_y(t) = dy/dt = [1407 * cos(44.7) - 50*9.81)/50] * t
y(t) = (1/2) [1407 * cos(44.7) - 50*9.81)/50] * t^2

Answer 2

Rocket Trajectory Equations

Answer 3

Force = mass x accelaration
accelaration=F/m
accelaration = dv/dt
velocity=integ. a .dt
s=ut+1/2at^2
where v is the velocity, s=distance travelled, u is the initial velocity and t the time.
Substituting this in S we get the distance at instance of time. Plot this distace against time and you get the trajectory.