At a time when mining asteroids has become feasible, astronauts have connected a line between their 3580-kg space tug and a 6000-kg asteroid. Using their ship's engine, they pull on the asteroid with a force of 490 N. Initially the tug and the asteroid are at rest, 400 m apart. How much time does it take for the ship and the asteroid to meet?
The center of mass will not accelerate, no external force on the system.
That is a distance d from the 6000 kg asteroid
6000 d = 3580(400 -d)
6000 d = 1,432,000 -3580 d
d = 149.5 meters from the asteroid
so we need to pull the asteroid 149.5 meters
149.5 = (1/2) a t^2
where a = F/m = 490/6000
149.5 = 245/6000 t^2
t = 60.5 seconds
about a minute
To find the time it takes for the ship and the asteroid to meet, we can use Newton's second law of motion, which states that the force exerted on an object is equal to the mass of the object multiplied by its acceleration:
F = m * a
In this case, the force exerted by the ship's engine pulls the asteroid towards the ship, causing it to accelerate. We can rearrange the formula to solve for acceleration:
a = F / m
Substituting the values given in the problem, we have:
a = 490 N / 6000 kg
Now, we need to find the acceleration of the ship. To do this, we can use the conservation of momentum. The initial momentum of the ship and the asteroid is zero since they are at rest. When they meet, their total momentum will still be zero. So we have:
momentum of the ship = - momentum of the asteroid
(mass of ship) * (acceleration of ship) = -(mass of asteroid) * (acceleration of asteroid)
(3580 kg) * (acceleration of ship) = -(6000 kg) * (acceleration of asteroid)
But since the ship and the asteroid are connected by a line, their accelerations will always be equal in magnitude but opposite in direction. So we can rewrite the equation as:
(3580 kg) * (acceleration) = (6000 kg) * (acceleration)
Now we can substitute for the acceleration:
(3580 kg) * (490 N / 6000 kg) = (6000 kg) * (acceleration)
Simplifying:
(3580 kg) * (490 N) = (6000 kg) * (acceleration)
Using the equation: F = m * a, we can solve for acceleration:
acceleration = F / m = (3580 kg * 490 N) / (6000 kg)
Now, we have the acceleration. Next, we can determine how long it takes for the ship and the asteroid to meet. We can use one of the equations of motion, namely:
s = ut + (1/2) * a * t^2
where:
s = distance traveled
u = initial velocity (which is 0 in this case since both are at rest)
a = acceleration
t = time
The ship is pulling the asteroid towards it, so the distance traveled, s, is 400 m. The acceleration, a, is what we calculated earlier. We want to find the time, t. Rearranging the equation, we get:
t = sqrt((2s) / a)
Substituting the values:
t = sqrt((2 * 400 m) / (acceleration))
Finally, we can calculate the time it takes for the ship and the asteroid to meet using the values we have found.