Question

A spaceship of mass 2.0 $\times$ 10⁶ kg is cruising at a speed of 5.0 $\times$ 10⁶ m/s when the antimatter reactor fails, blowing the ship into three pieces. One section, having a mass of 5.0 $\times$ 10⁵ kg, is blown straight backward with a speed of 2.0 $\times$ 10⁶ m/s. A second piece, with mass 8.0 $\times$ 10⁵ kg, continues forward at 1.0 $\times$10⁶ m/s. What are the direction and speed of the third piece?

Short answer

Velocity of the third piece is 14.6$\times$10⁶ms^-1in forward direction.

Step-by-step solution

Step 1. Given information isM = 2 × 106 kgm1 = 5 × 105 kgm2 = 5 × 105 kgm3 = mgvs = 5 × 106 ms-1v1  = 2 × 106 ms-1v2 = 1 × 106 ms-1

We need to find the direction and speed of the third piece .

Step 2. Using the law of conservation of mass, we will find out the mass of third piece.

The spaceship initially has a mass M = 2 x 10⁶ kg. After the explosion, the spaceship splits into three pieces,

Using conservation of momentum,

M = m₁ + m₂ + m₃

m₃ = M - m₁- m₂

m₃= (2$\times$10⁶ kg) - (5$\times$10⁵ kg) - (8$\times$10⁵ kg)

m₃ = 7$\times$10⁵kg

= 0.7$\times$10⁶ kg

Step 3. When an object splits into sections, the explosion between them is an inelastic collision where momentums is conserved. The objects have the same initial velocity and we will use the law of conservation of momentum to find out the velocity of third piece.

As per law of conservation of momentum

$\overrightarrow{P_i}=\overrightarrow{P_f}$

And momentum is expressed as

P = mv

Therefore,

M_sv_s = m₁(-v₁) + m₂v₂ + m₃v₃

v₁is taken as negative because it is moving in backward direction.

Putting values,

localid="1648120532677" $2\times {10}^6\times 5\times {10}^{6}kgm{s}^{-1}=-(0.5\times {10}^6\times 2\times {10}^{6}kgm{s}^{-1})+(0.8\times {10}^6\times 1\times {10}^{6}kgm{s}^{-1})+(0.7\times {10}^{6}kg\times v_3)$

v₃ =localid="1648120466371" $\frac{(10+0.2)\times {10}^{12}kgm{s}^{-1}}{0.7\times {10}^{6}kg}=\frac{10.2}{0.7}\times {10}^{6}m{s}^{-1}=14.6\times {10}^{6}m{s}^{-1}$

As the value of v₃ is positive, it will move in forward direction.