For Example 8.1, what initial launch angle should be chosen to maximize the horizontal dis...

For Example 8.1, what initial launch angle should be chosen to maximize the horizontal distance traversed?

Example 8.1

A popular college project is to have students build model rockets and predict their performance, and then launch the rockets and see how close the actual performance comes to the predicted. Suppose you have built a model rocket of mass 50 g, length 30 cm, tube diameter of 2.5 cm, and drag coefficient of 0.75, to be used with a small motor that produces a total of 4 N of thrust over 0.5 s of firing lime. If the rocket is launched at an angle of 60° to the horizontal, how far will it travel horizontally? Assume there is no significant wind at the time of launch.

SOLUTION To solve this problem, we could use Algorithm 6.1, but it must be modified to include a thrust force. This is done in Algorithm 8.1. For these conditions, the numerical solution calculated in MATLAB is that the rocket will travel 110 m horizontally, with a maximum altitude of 15 m.

ALGORITHM 8.1

1. Set the initial conditions x0, y0, Vx0, Vy0. If the magnitude, V0, and angle, θ, of the initial velocity are specified instead, then Vx0 = V0 cos θ and Vy0 = V0 sin (θ)

2. Set x(0) = x0, y(0) = y0, Vx(0) = Vx0, Vy(0) = Vy0.

3. Set the initial time t = 0, and select the numerical time step, Δt.

4. Specify the mass, m, and cross-sectional area, A, of the projectile and the density, ρ, and viscosity, v, of air for local atmospheric conditions. If the rocket is going to high altitudes, include a correlation or look-up table for density as a function of height.

5. Specify the thrust curve FT(t) for the rocket motor.

6. LOOP: For the current time value t

Compute the magnitude of velocity: Compute the Reynolds number: Re = Vd/v

Compute the drag coefficient from the appropriate correlation: CD = CD(Re)

Compute the drag force: F =  CD ρAV2

Compute the x and y components of the drag force:

Fx = F * Vx(t)/V
Fy = F * Vy(t)/ V

Look up the magnitude of the thrust force from the stored thrust curve:

Ft = FT(t)

Compute the x and y components of the thrust force:

Ft,x = Ft * Vx(t)/V
FT,y = Ft * Vy(t)/V

Perform integrations of the four variables:

vx(t + Δt) = Vx(t) + (−FD,X + FT,X)/m * Δt
Vy(t + Δt) = Vy(t) + (−9 − FD,y/m + FT,y/m) * Δt
x(t + Δt) = x(t) + Vx(t) * Δt
y(t + Δt) = y(t) + Vy(t) * Δt

Increment the current time by time step: t = t + Δt

7. END LOOP when y < 0