Answer :
To solve the problem of finding the astronaut's recoil velocity when she throws a wrench in space, we can apply the principle of conservation of momentum. This principle states that the total momentum of an isolated system remains constant if no external forces act upon it.
Here's a step-by-step solution:
1. Identify the Initial Condition:
- The astronaut and the wrench are initially at rest in space. Therefore, the total initial momentum of the system is zero.
2. Set Up the Equation Using Conservation of Momentum:
- After the astronaut throws the wrench, the total momentum of the system must still be zero.
- The equation for conservation of momentum is given by:
[tex]\[
\text{Initial Momentum} = \text{Final Momentum}
\][/tex]
[tex]\[
0 = (\text{mass of astronaut} \times \text{velocity of astronaut}) + (\text{mass of wrench} \times \text{velocity of wrench})
\][/tex]
3. Plug in the Given Values:
- Mass of the astronaut = 143 kg
- Mass of the wrench = 0.725 kg
- Velocity of the wrench (thrown forward) = 13.8 m/s
4. Calculate the Recoil Velocity of the Astronaut:
- Since the initial momentum is zero, the final momentum must also equate to zero:
[tex]\[
0 = (143 \times \text{velocity of astronaut}) + (0.725 \times 13.8)
\][/tex]
- Rearrange the equation to solve for the velocity of the astronaut:
[tex]\[
\text{velocity of astronaut} = -\frac{0.725 \times 13.8}{143}
\][/tex]
5. Result:
- The recoil velocity of the astronaut is approximately [tex]\(-0.07 \, \text{m/s}\)[/tex].
The negative sign indicates that the astronaut moves in the opposite direction to the wrench's movement, since she is recoiling.
Here's a step-by-step solution:
1. Identify the Initial Condition:
- The astronaut and the wrench are initially at rest in space. Therefore, the total initial momentum of the system is zero.
2. Set Up the Equation Using Conservation of Momentum:
- After the astronaut throws the wrench, the total momentum of the system must still be zero.
- The equation for conservation of momentum is given by:
[tex]\[
\text{Initial Momentum} = \text{Final Momentum}
\][/tex]
[tex]\[
0 = (\text{mass of astronaut} \times \text{velocity of astronaut}) + (\text{mass of wrench} \times \text{velocity of wrench})
\][/tex]
3. Plug in the Given Values:
- Mass of the astronaut = 143 kg
- Mass of the wrench = 0.725 kg
- Velocity of the wrench (thrown forward) = 13.8 m/s
4. Calculate the Recoil Velocity of the Astronaut:
- Since the initial momentum is zero, the final momentum must also equate to zero:
[tex]\[
0 = (143 \times \text{velocity of astronaut}) + (0.725 \times 13.8)
\][/tex]
- Rearrange the equation to solve for the velocity of the astronaut:
[tex]\[
\text{velocity of astronaut} = -\frac{0.725 \times 13.8}{143}
\][/tex]
5. Result:
- The recoil velocity of the astronaut is approximately [tex]\(-0.07 \, \text{m/s}\)[/tex].
The negative sign indicates that the astronaut moves in the opposite direction to the wrench's movement, since she is recoiling.
