Question

A small spaceship with a mass of only${\mathbf{\text{1.5×10}}}^{\mathbf{\text{3}}}\mathbf{\text{kg}}$ (including an astronaut) is drifting in outer space with negligible gravitational forces acting on it. If the astronaut turns on a 10kw laser beam, what speed will the ship attain in 1.0 day because of the momentum carried away by the beam.

Short answer

The speed that the ship will attain in 1.0 day because of the momentum carried away by the beam is $v=1.9\times {10}^{-3}\text{\hspace{0.17em}m/s}$

Step-by-step solution

Step 1: Given

$\begin{array}{l}m=1500\text{\hspace{0.17em}kg;}\\ P=\text{10kW=10000W}\\ t=1\text{day=86400s}\end{array}$

Determining the concept

The force can be found by using the relation between power, force, and velocity. Then, by using the formulae of momentum find the unknown speed of the ship. It has a magnitude and a direction. The direction towards which the force is applied is known as the direction of the force and the point where force is applied is the point of action of force. The SI unit of force is Newton (N).

Power (P):

$P=Fv$

Where F is force and v is speed

Momentum (p),

P=mv

=Fv

where, p is the momentum, m is the mass.

(a) Determining the speed will the ship attain in 1.0 day because of the momentum carried away by the beam

Power (P) is given by,

P=Fv

Here, v= speed of light

$\left(10000\text{\hspace{0.17em}W}\right)=F\times (3\times {10}^8\text{\hspace{0.17em}m}/\text{s})$

Hence,

$F=333.33\times {10}^{-7}\text{N}$

By using the momentum equations:

Ft = mv

Here, v= speed of ship

$\begin{array}{l}333.33\times {10}^{-8}\text{\hspace{0.17em}N}\times 86400\text{\hspace{0.17em}s}=1500\times v\\ v=\frac{333.33\times {10}^{-7}\text{\hspace{0.17em}N}\times 86400\text{\hspace{0.17em}s}}{1500\text{\hspace{0.17em}kg}}\\ v=1.9\times {10}^{-3}\text{\hspace{0.17em}m/s}\end{array}$

Therefore, the speed that the ship will attain in 1.0 day because of the momentum carried away by the beam is$v=1.9\times {10}^{-3}\text{\hspace{0.17em}m/s}$