Answer to Question #86672 in Mechanics | Relativity for Jon

Question

A rocket, with a mass of 1800 kg, accelerates upwards at 85.0 m/s/s. Unfortunately, it explodes 2.5 s after take off into two different size pieces. One piece is two times as massive as the other piece. Radar evidence showed the smaller piece traveled at an angle of 10.0o from the original trajectory of the rocket, in a westerly direction, to a height of 9000 m. The larger piece was not recorded and saved by radar as a result of a malfunction.
a). Determine the speed and direction of the larger, two times as massive, piece?
b). How far, and in what direction from the launch site, should the search time look to find the larger piece?

Answer 1

Given:

$M = 1800~\text{kg};~a = 85.0~\frac{\text{m}}{\text{s}^2};~t = 2.5~\text{s}; \\ m_1 = 2m_2;~{\alpha}_2=10^\circ;~h_2 = 9000~\text{m}. \\$

First of all, let's note that we can determine the height and speed (naturally upwards vertical) of the intact rocket just at the moment of the explosion as:

$h = \frac{1}{2}at^2;~v = at.~$

Let's also remember that we can determine the speed of the smaller piece (just after the explosion), making use of the equation for height of its ballistic trajectory:

$h_2 = h + \frac{{v_2}^2{\cos^2{\alpha}_2}}{2g}; \implies v_2 = \frac{\sqrt{g(2h_2-at^2)}}{\cos\alpha_2}.$

a)

Determine the speed and direction of the larger, two times as massive, piece?

We'll assume that the larger piece directed at the east-upwards, at an angle ${\alpha}_1~$between its direction and the upwards vertical (original rocket direction), at a speed $v_1~$immediately after the explosion.

Let's use the conservation of momentum just before and after the explosion:

$\begin{cases} 0 = m_1v_1\sin{\alpha}_1 - m_2v_2\sin{\alpha}_2; \\ Mv = m_2v_2\cos{\alpha}_2 + m_1v_1\cos{\alpha}_1. \end{cases}$

Solving it for the unknowns,

$\tan\alpha_1 = \frac{m_2v_2\sin\alpha_2}{m_2v_2\cos\alpha_2-Mv} = \tan\alpha_2\frac{m_2\sqrt{g(2h_2-at^2)}}{m_2\sqrt{g(2h_2-at^2)}-Mat}; \\$

$v_1 = \frac{1}{m_1}\sqrt{\frac{m_2^2g(2h_2-at^2)}{\cos^2\alpha_2} + (Mat)^2 - 2Mm_2at\sqrt{g(2h_2-at^2)}}; \\$

We can simplify those a bit by canceling masses if we recall that $m_1 = 2m_2; M = 3m_2.$

$v_1 = \frac{1}{2}\sqrt{\frac{g(2h_2-at^2)}{\cos^2\alpha_2} + (3at)^2 - 6at\sqrt{g(2h_2-at^2)}}; \\$

$\alpha_1 = \arctan{\lbrack\tan(\alpha_2)\frac{\sqrt{g(2h_2-at^2)}}{3at-\sqrt{g(2h_2-at^2)}}\rbrack}; \\$

With numerical values,

$v_1 \approx 117.57~\frac{\text{m}}{\text{s}}; \\ \alpha_1 \approx 18.08^\circ. \\$

b)

In order to find the horizontal range of the larger piece we will have to write its kinematic equations for horizontal and vertical axes, using the time reference $\tau~$ from the moment of the explosion:

$\begin{cases} x_1 = v_1\sin\alpha_1\tau; \\ 0 = h + v_1\cos\alpha_1\tau - \frac{1}{2}g\tau^2. \\ \end{cases}$

The last equation shows the height 0 at the moment of eventual falling.

Rewriting it in form of quadratic equation and replacing h, we get and solve:

$g\tau^2 - 2v_1\cos{(\alpha_1)}\tau - at^2 = 0; \\ \tau_{i,j} = \frac{v_1\cos\alpha_1 \pm \sqrt{v_1^2\cos^2\alpha_1 + gat^2}}{g}; \\$

Obviously we need the positive solution as the negative one corresponds to the wrong direction of time and, correspondingly, direction of movement.

As the formulas would become too cumbersome if were to express the values through the given original values, we will use some intermediate calculations:

$\tau \approx 24.96~\text{s}; \\ x_1 = 117.57~\frac{\text{m}}{\text{s}}~*~\sin{(18.08^\circ)}~*~24.96~\text{s} \approx 910.82~\text{m}.$

The rescue team should look in east direction from the launch site.