Question

A spacecraft of mass 25 t flies in infinite space and under the action of constant reactive force traverses 2000 km of path for 30 min. Calculate the reactive force, when it is known that at the moment of starting the engines the speed of the ship (in relation to the center of the Galaxy) was Vo = 1500 km / s.

Answer 1

First, let's calculate the acceleration, caused by the reactive force. The distance traveled under the constant acceleration is:

$$d = v_0t+\dfrac{at^2}{2}$$

where $v_0 = 1500km/s = 1.5\times 10^6m/s$ is the initial speed, $t = 30min = 1.8\times 10^3s$ is the time of force acting, and $a$ is the acceleration. Since $d = 2000km = 2\times 10^6m$, obtain:

$$a = \dfrac{2d}{t^2} - \dfrac{2v_0}{t}$$

The projection of the reactive force on the line of motion is (according to the second Newton's law):

$$F = ma$$

where $m = 25t = 25\times 10^6 kg$ is the mass of the spacecraft. Finally, obtain:

$$F_x = 2m\left( \dfrac{d}{t^2} - \dfrac{v_0}{t} \right)\\ F = 2\cdot 25\times 10^6\cdot \left( \dfrac{2\times 10^6}{(1.8\times 10^3)^2} - \dfrac{1.5\times 10^6}{1.8\times 10^3} \right) \approx -4.16 \times 10^{10}N$$

The magnitude of the force is:

$$F = 4.16 \times 10^{10}N$$

Answer. $F = 4.16 \times 10^{10}N$.