At a time when mining asteroids has become feasible, astronauts have connected a line between their 3830-kg space tug and a 5930-kg asteroid. They pull on the asteroid with a force of 671 N. Initially the tug and the asteroid are at rest, 514 m apart. How much time does it take for the ship and the asteroid to meet?
the tug and the asteroid are accelerated towards each other by the tension in the line
f = m a ... a = f / m
a1 = 671 / 3830
a2 = 671 / 5930
d1 + d2 = 514
1/2 a1 t^2 + 1/2 a2 t^2 = 514
(a1 + a2) t^2 = 1028
To find the time it takes for the ship and the asteroid to meet, we can use the equation of motion:
F = ma
Where:
F is the force applied (671 N)
m is the total mass (3830 kg + 5930 kg = 9760 kg)
a is the acceleration
We can rearrange the equation to solve for acceleration:
a = F / m
Plugging in the given values:
a = 671 N / 9760 kg
Next, we can use the equation of motion again to find the time it takes for the asteroid and the ship to meet:
d = v₀t + 0.5at²
Where:
d is the initial distance between the asteroid and the ship (514 m)
v₀ is the initial velocity (0 m/s)
t is the time
a is the acceleration
Since the asteroid is initially at rest, its initial velocity is 0 m/s.
The equation simplifies to:
d = 0.5at²
Rearranging the equation to solve for time:
t² = (2d) / a
Plugging in the known values:
t² = (2 * 514 m) / (671 N / 9760 kg)
Now we can calculate the time it takes for the ship and the asteroid to meet by taking the square root of both sides of the equation:
t = √[(2 * 514 m) / (671 N / 9760 kg)]
Evaluating the expression:
t ≈ 11.8 seconds
Therefore, it takes approximately 11.8 seconds for the ship and the asteroid to meet.