At a time when mining asteroids has become feasible, astronauts have connected a line between their 3140-kg space tug and a 5050-kg asteroid. They pull on the asteroid with a force of 589 N. Initially the tug and the asteroid are at rest, 504 m apart. How much time does it take for the ship and the asteroid to meet?
for each body, F=ma
So, get the acceleration, and then using
s = 1/2 at^2
add the two distances and find t when their sum is 504m
To determine the time it takes for the ship and the asteroid to meet, we can use Newton's second law of motion. First, let's calculate the acceleration of the asteroid using the force applied.
Newton's second law states: force (F) equals mass (m) multiplied by acceleration (a), or F = ma.
We know the force applied (F) is 589 N, and the mass of the asteroid (m) is 5050 kg. Rearranging the formula to solve for acceleration, we get:
a = F / m
Plugging in the given values, we have:
a = 589 N / 5050 kg
a ≈ 0.11683 m/s^2 (rounded to 5 decimal places)
Now, let's find the time it takes for the ship and asteroid to meet, assuming they start from rest and are initially 504 m apart. We can use the formula:
s = ut + (1/2)at^2
where:
- s is the distance traveled,
- u is the initial velocity (which is 0 m/s in this case),
- a is the acceleration, and
- t is the time taken.
We rearrange the formula to solve for time (t):
(1/2)at^2 = s
t^2 = (2s) / a
t = sqrt((2s) / a)
Plugging in the values, we have:
t = sqrt((2 * 504 m) / 0.11683 m/s^2)
t ≈ sqrt(8641.53) s
t ≈ 93.01 s
Therefore, it takes approximately 93.01 seconds for the ship and the asteroid to meet.