A(n) 57.3 kg astronaut becomes separated

from the shuttle, while on a space walk. She finds herself 54.3 m away from the shuttle and moving with zero speed relative to the shuttle. She has a(n) 0.697 kg camera in her hand and decides to get back to the shuttle
by throwing the camera at a speed of 12m/s in the direction away from the shuttle. How long will it take for her to reach the shuttle?
Answer in minutes.
Answer in units of min.

4.5 min

To solve this problem, we can use the principle of conservation of momentum.

Step 1: Calculate the astronaut's initial momentum.
The initial momentum of the astronaut and the camera is given by:

p_initial = (mass_astronaut + mass_camera) * velocity_astronaut

Given:
mass_astronaut = 57.3 kg
mass_camera = 0.697 kg
velocity_astronaut = 0 m/s (zero speed relative to the shuttle)

p_initial = (57.3 kg + 0.697 kg) * 0 m/s
p_initial = 0 kg*m/s

Step 2: Calculate the momentum of the astronaut and camera after throwing the camera.
The final momentum of the astronaut and camera is given by:

p_final = mass_astronaut * velocity_astronaut_final

Given:
velocity_astronaut_final = -12 m/s (negative sign indicates opposite direction)

p_final = 57.3 kg * (-12 m/s)
p_final = -687.6 kg*m/s

Step 3: Apply the conservation of momentum principle.
According to the principle of conservation of momentum, the total momentum before and after throwing the camera should be equal:

p_initial = p_final
0 kg*m/s = -687.6 kg*m/s

Step 4: Solve for the time it takes for the astronaut to reach the shuttle.
We can calculate the time using the equation:

t = distance / velocity_astronaut_final

Given:
distance = 54.3 m
velocity_astronaut_final = -12 m/s

t = 54.3 m / (-12 m/s)
t ≈ -4.525 s

However, time cannot be negative in this case because we are interested in the time it takes for the astronaut to reach the shuttle. Therefore, the negative sign simply indicates the direction. We can take the absolute value of the time:

t ≈ 4.525 s ≈ 4.525 seconds

Converting to minutes:

t_min = 4.525 s / 60
t_min ≈ 0.07542 minutes ≈ 0.075 min (rounded to 3 decimal places)

Therefore, it will take approximately 0.075 minutes for the astronaut to reach the shuttle.

To find the time it takes for the astronaut to reach the shuttle, we can use the concept of conservation of momentum.

First, let's calculate the initial momentum of the astronaut-camera system. The momentum (p) is given by the product of mass (m) and velocity (v):

Initial momentum of astronaut-camera system = (mass of astronaut) × (velocity of astronaut) + (mass of camera) × (velocity of camera)

The mass of the astronaut is given as 57.3 kg, and since she is at rest relative to the shuttle, her velocity is zero. The mass of the camera is given as 0.697 kg, and its initial velocity is 12 m/s. Putting these values into the equation, we get:

Initial momentum of astronaut-camera system = (57.3 kg) × (0 m/s) + (0.697 kg) × (12 m/s)
= 0 + 8.364 kg·m/s
= 8.364 kg·m/s

According to the conservation of momentum, the total momentum of the system remains constant unless acted upon by an external force. When the astronaut throws the camera, an external force is applied to the system.

Let's assume the astronaut and camera move towards the shuttle with a final velocity of v (unknown). Since the astronaut and camera are moving in opposite directions, their velocities add up algebraically, which gives us:

(mass of astronaut) × (final velocity of astronaut) + (mass of camera) × (final velocity of camera) = 8.364 kg·m/s

(57.3 kg) × (v) + (0.697 kg) × (-12 m/s) = 8.364 kg·m/s

Now we can solve for v:

57.3v - 8.364 = -0.697 × 12
57.3v = -8.364 + 8.364
57.3v = 8.364
v = 8.364 / 57.3
v ≈ 0.146 m/s

Now, we need to find the time it takes for the astronaut to reach the shuttle. We can use the equation for distance (d) traveled at a constant speed (v) over time (t):

d = v × t

Where d is the distance, v is the velocity, and t is the time.

In this case, the distance is given as 54.3 m, and the velocity is 0.146 m/s. Plugging these values into the equation gives us:

54.3 = 0.146 × t

Now, let's solve for t:

t = 54.3 / 0.146
t ≈ 372.61 s

Finally, we need to convert the time from seconds to minutes. There are 60 seconds in a minute, so:

t (in minutes) = 372.61 s / 60
t ≈ 6.21 min

Therefore, it will take approximately 6.21 minutes for the astronaut to reach the shuttle.