Question

A Federation starship (8.5 ✕ 106 kg) uses its tractor beam to pull a shuttlecraft (1.0 ✕ 104 kg) aboard from a distance of 14 km away. The tractor beam exerts a constant force of 4.0 ✕104 N on the shuttlecraft. Both spacecraft are initially at rest. How far does the starship move as it pulls the shuttlecraft aboard?

Answer 1

Refer to the diagram shown below.

m₁ = 8.5 x 10⁶ kg, the mass of the starship
m₂ = 10⁴ kg, the mass of the shuttlecraft
a₁ =  the acceleration of the starship
a₂ = the acceleration of the shuttle
F = 4 x 10⁴ N, the pulling force

Let y =  distance traveled by the starship
Let x =  distance traveled by the shuttlecraft
If t =  the time of travel, then
y = 0.5a₁t²                  (1)
x = 0.5a₂t²                  (2)

F = m₁a₁ = m₂a₂         (3)
Also,
x + y = 14000 m          (4)

From (2), obtain
a₁ = (4 x 10⁴ N)/(8.5 x 10⁶ kg) = 4.706 x 10⁻³ m/s²
a₂ = (4 x 10⁴ N)/(10⁴ kg) = 4 m/s²

From (1), (2) and (4), obtain
0.5*(t s)²*(4 + 4.706 x 10⁻³ m/s²) = 14000 m
2.002353t² = 14000
t² = 6991.774 s²
t = 83.617 s

Therefore
x = 0.5*4*6991.774 = 13984 m = 13.984 km
y = 0.5*4.706 x 10⁻³*6991.774 = 16.452 m

The starship moves only about 16.5 m while pulling in the shuttlecraft by 13.98 km.

Answer: The starship moves by 16.5 m (nearest tenth)