At a time when mining asteroids has become feasible, astronauts have connected a line between their 3760-kg space tug and a 6100-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, 490 m apart. How much time does it take for the ship and the asteroid to meet
To calculate the time it takes for the ship and the asteroid to meet, we need to use Newton's second law of motion. The equation is:
Force = Mass x Acceleration
First, let's find the acceleration of the system. Since the only force acting on the system is the force applied by the ship's engine, we can find the acceleration using the equation:
Force = Mass x Acceleration
Rearranging the equation to solve for acceleration:
Acceleration = Force / Mass
For the ship:
Mass of the ship (m_1) = 3760 kg
For the asteroid:
Mass of the asteroid (m_2) = 6100 kg
Force applied by the ship (F) = 490 N
Now, we can calculate the acceleration:
Acceleration = Force / (Mass of ship + Mass of asteroid)
Acceleration = 490 N / (3760 kg + 6100 kg)
Acceleration = 490 N / 9860 kg
Next, we can use the kinematic equation to find the time it takes for the ship and the asteroid to meet. The equation is:
Distance = Initial Velocity x Time + 0.5 x Acceleration x Time^2
In this case, the distance (s) is 490 m, the initial velocity (v) is 0 m/s since both the ship and asteroid are at rest initially, and the acceleration (a) is the value we calculated earlier.
490 m = 0 x Time + 0.5 x (490 N / 9860 kg) x Time^2
Simplifying the equation:
245 m = (245 N / 9860 kg) x Time^2
Rearranging the equation to solve for time:
Time^2 = (245 m x 9860 kg) / 245 N
Time^2 = 9860 kg
Taking the square root of both sides:
Time = √9860 kg
Calculating the time:
Time = 31.39 seconds
Therefore, it takes approximately 31.39 seconds for the ship and the asteroid to meet.