Question 5.3.2: A Rocket-Propelled Sled A rocket-propelled sled on a track i...
A Rocket-Propelled Sled
A rocket-propelled sled on a track is represented in Figure 5.3.2 as a mass m with an applied force f that represents the rocket thrust. The rocket thrust initially is horizontal, but the engine accidentally pivots during firing and rotates with an angular acceleration of \ddot{θ} = \pi/50 rad/s.
Compute the sled’s velocity v for 0 ≤ t ≤ 6 if v(0) = 0. The rocket thrust is 4000 N and the sled mass is 450 kg.
The sled’s equation of motion is 450\dot{v} = 4000 \cos θ(t) . To obtain θ(t), note that
\dot{θ} = \int^{t}_{0} {\ddot{θ} dt} = \frac{\pi}{50} t
and
θ = \int^{t}_{0} {\dot{θ} dt} = \int^{t}_{0} {\frac{\pi}{50} t dt} = \frac{\pi}{100} t^{2}
Thus the equation of motion becomes
\dot{v} = \frac{80}{9} \cos \left(\frac{\pi}{100} t^{2} \right) (1)
The solution is formally given by
v(t) = \frac{80}{9} \int^{t}_{0} {\cos \left(\frac{\pi}{100} t^{2} \right) dt}
Unfortunately, no closed-form solution is available for the integral, which is called Fresnel’s cosine integral. The value of the integral has been tabulated numerically, but we will use a MATLAB ODE solver to obtain the solution.
First create the following user-defined function file, which is based on equation (1).
function vdot = sled(t,v)
vdot = 80*cos(pi*t^2/100)/9;
As a check on our results, we will use the solution for θ = 0 as a comparison. The equation of motion for this case is \dot{v} = 80/9, which gives v(t) = 80t/9. The following session solves the equation and plots the two solutions, which are shown in Figure 5.3.3.
[t,v] = ode45(@sled,[0 6],0);
plot(t,v,t,(80*t/9)),xlabel('t (s)'),...
ylabel('v (m/s)'),gtext('\theta = 0'),gtext('\theta \neq 0')
We can make two observations that help us determine whether or not our numerical solution is correct. From the plot we see that the solution for θ ≠ 0 is almost identical to the solution for θ = 0, for small values of θ. This is correct because \cos θ \approx 1 for small values of θ. As θ increases, we would expect the velocity to be smaller than the velocity for θ = 0 because the horizontal component of the thrust is smaller. The plot confirms this.
