A 200.0 kg astronaut and equipment move with a velocity of 2.00 m/s toward an orbiting

spacecraft. The astronaut will fire a 100.0 N rocket backpack to stop his motion relative to
the spacecraft. (a) What acceleration is attained by the rocket backpack?

(a) a = F/m = 100/200 = 0.5 m/s^2

To solve this problem, we can use Newton's second law of motion, which states that the force acting on an object is equal to its mass multiplied by its acceleration (F = ma).

Given:
Mass of the astronaut and equipment (m) = 200.0 kg
Velocity of the astronaut and equipment (v) = 2.00 m/s
Force applied by the rocket backpack (F) = 100.0 N

We need to find the acceleration of the astronaut and equipment (a).

Since the rocket backpack is applied in the opposite direction to the motion of the astronaut and equipment, the net force acting on them will be the difference between the force applied by the rocket backpack and the force required to move the astronaut and equipment. The force required to move the astronaut and equipment is equal to the product of their mass and acceleration.

The equation for net force is:
Net force = Force applied by the rocket backpack - Force required to move the astronaut and equipment

We can express the force required to move the astronaut and equipment using Newton's second law:
Force required to move the astronaut and equipment = mass x acceleration

Substituting the given values into the equation, we have:
Net force = 100.0 N - (200.0 kg x acceleration)

Since the net force is equal to the mass times the acceleration, we can write the equation as:
100.0 N - (200.0 kg x acceleration) = 200.0 kg x acceleration

Now, we can solve for acceleration:
100.0 N = 400.0 kg x acceleration
acceleration = 100.0 N / 400.0 kg

Therefore, the acceleration attained by the rocket backpack is 0.25 m/s^2.