A 88-kg astronaut and a 1200-kg satellite are at rest relative to the space shuttle. The astronaut pushes on the satellite, giving it a speed of 0.25m/s directly away from the shuttle. Seven-and-a-half seconds later the astronaut comes into contact with the shuttle.What was the initial distance from the shuttle to the astronaut?.

Astronaut's speed toward shuttle after pushing satellite can be computed by assuming total momentum remains zero. Satellite and astronaut momentum with respect to shuttle are equal and opposite, so

V = 1200*0.25/88 = 3.41 m/s

Initial distance apart = V*7.5 s = 25.6 m

To find the initial distance from the shuttle to the astronaut, we can use the equation of motion:

d = v * t

where:
d = distance
v = velocity
t = time

Given that the astronaut pushed the satellite with a speed of 0.25 m/s directly away from the shuttle and that the astronaut came into contact with the shuttle 7.5 seconds later, we can calculate the initial distance.

Let's substitute the given values into the equation:

d = 0.25 m/s * 7.5 s

d = 1.875 meters

Therefore, the initial distance from the shuttle to the astronaut was 1.875 meters.

To find the initial distance from the shuttle to the astronaut, you can use the concept of relative motion.

First, let's calculate the momentum of the satellite after it is pushed by the astronaut.

The momentum (p) of an object can be calculated by multiplying its mass (m) by its velocity (v).
For the satellite:
Mass (m) = 1200 kg
Velocity (v) = 0.25 m/s

Momentum (p) = mass (m) × velocity (v)
p = 1200 kg × 0.25 m/s
p = 300 kg⋅m/s

According to the law of conservation of momentum, the total momentum before and after the interaction should be the same.

Since the astronaut and the shuttle were initially at rest relative to each other, the initial momentum of the system is zero.

Therefore, the momentum of the astronaut after coming into contact with the shuttle should be equal in magnitude but opposite in direction to the momentum of the satellite.

Momentum of the astronaut (p_astronaut) = -300 kg⋅m/s

Now, let's calculate the momentum of the astronaut using the following formula:

Momentum (p_astronaut) = mass (m_astronaut) × velocity (v_astronaut)

Given that the mass of the astronaut (m_astronaut) is 88 kg, we can rearrange the equation to solve for the velocity (v_astronaut).

v_astronaut = p_astronaut / m_astronaut
v_astronaut = -300 kg⋅m/s / 88 kg
v_astronaut ≈ -3.41 m/s

The negative sign indicates that the astronaut is moving in the opposite direction of the satellite.

Now, we can calculate the distance traveled by the astronaut during the 7.5 seconds using the formula:

Distance (s) = velocity (v_astronaut) × time (t)
s = -3.41 m/s × 7.5 s
s ≈ -25.59 m

Since distance cannot be negative in this context, we take the magnitude of the distance.

Distance (s) = |-25.59 m|
s ≈ 25.59 m

Therefore, the initial distance from the shuttle to the astronaut was approximately 25.59 meters.