Answer:
She can be up to 110 m from the craft and still return safely.
Step-by-step explanation:
Hi there!
To solve this problem, we have to apply the conservation of momentum. The momentum of the system astronaut-oxygen tank remains constant. Since initially, the system is at rest, the initial momentum is zero, and, since it is conserved (i.e. it remains constant), the final momentum will also be zero.
The final momentum is calculated as follows:
Final momentum = p tank + p astronaut
Where:
p tank = momentum of the tank
p astronaut = momentum of the astronaut
The momentum of each component of the system is calculated as follows:
p = m · v
Where:
m = mass of the object.
v = velocity.
Then, if:
mt = mass of the tank.
vt = velocity of the tank.
ma = mass of the astronaut.
va = velocity of the astronaut
and we consider the direction towards the shuttle as the positive direction, the momentum of the system can be expressed as follows:
final momentum = ma · va - mt · vt
0 = 64.7 kg · va - 14.1 kg · 7.45 m/s
Solving for va:
va = 14.1 kg · 7.45 m/s / 64.7 kg
va = 1.62 m/s
The astronaut will travel at 1.62 m/s towards the shuttle. Let´s calculate how much distance can she travel in 68.2 s using this equation:
x = x0 + v0 · t + 1/2 · a · t²
Where:
x = traveled distance at a time "t".
x0 = initial position.
v0 = initial velocity.
t = time.
a = acceleration.
Since there is no force acting on the astronaut, the acceleration will be zero (a = 0). If we place the origin of the frame of reference at the point where she throws the tank, x0 = 0. Then, the equation will be:
x = v · t
x = 1.62 m/s · 68.2 s
x = 110 m
She can be up to 110 m from the craft and still return safely.