At a time when mining asteroids has become feasible, astronauts have connected a line between their 3640 kg space tug and a 6500 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, 500 m apart. How much time does it take for the ship and the asteroid to meet?
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 applied on an object is equal to the mass of the object multiplied by its acceleration.
First, we need to calculate the acceleration of the combined system (the ship and the asteroid). We can do this by using the total force applied and the total mass of the system.
The total mass of the system is the sum of the mass of the ship and the mass of the asteroid:
Total mass = mass of the ship + mass of the asteroid
Total mass = 3640 kg + 6500 kg
Total mass = 10140 kg
Next, we can calculate the acceleration of the system using Newton's second law:
Force = mass × acceleration
Acceleration = Force / mass
Acceleration = 490 N / 10140 kg
Acceleration ≈ 0.0483 m/s^2
Now, we can use the equations of motion to find the time it takes for the ship and the asteroid to meet. Since the initial velocity of both the ship and the asteroid is zero, we can use the equation:
Distance = (1/2) × acceleration × time^2
Rearranging the equation to solve for time:
time = sqrt((2 × distance) / acceleration)
Plugging in the given values:
distance = 500 m
acceleration ≈ 0.0483 m/s^2
time = sqrt((2 × 500 m) / 0.0483 m/s^2)
time ≈ sqrt(20618.13)
time ≈ 143.60 seconds
Therefore, it will take approximately 143.60 seconds for the ship and the asteroid to meet.
To determine 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 applied to an object is equal to the mass of the object multiplied by its acceleration. In this case, the force applied is 490 N, and we need to find the acceleration of the asteroid.
To find the acceleration, we can use the equation F = ma, where F is the force applied, m is the mass, and a is the acceleration. Rearranging the equation, we have a = F/m.
The total mass of the system (tug + asteroid) is the sum of their individual masses:
Total mass = Mass of the tug + Mass of the asteroid
Total mass = 3640 kg + 6500 kg
Total mass = 10140 kg
Now we can calculate the acceleration:
a = F/m = 490 N / 10140 kg
The force is acting in the direction of motion, so the acceleration is positive:
a = 0.0482 m/s²
Now that we have the acceleration, we can use the equations of motion to determine the time it takes for the ship and the asteroid to meet. The equation we will use is:
s = ut + (1/2)at²
Where:
s = distance traveled by the asteroid (500 m)
u = initial velocity of the asteroid (0 m/s)
a = acceleration of the asteroid (0.0482 m/s²)
t = time
Plugging in the values, we can solve for t:
500 m = 0 m/s * t + (1/2)(0.0482 m/s²)(t²)
500 m = (0.0241 m/s²)(t²)
t² = 500 m / 0.0241 m/s²
t² ≈ 20747.71 s²
t ≈ √20747.71
t ≈ 144 s
Therefore, it takes approximately 144 seconds for the ship and the asteroid to meet.