To solve this problem, let's analyze the forces acting on the rocket and use the work-energy principle. The problem asks us to find the rocket's speed after traveling 37.0 cm (0.37 m) upward without the influence of the spring.

Given Data

• Mass of the rocket, $m = 10.9 \, \text{kg}$
• Thrust force generated by the rocket, $F_{\text{thrust}} = 210 \, \text{N}$
• Gravitational force acting on the rocket, $F_{\text{gravity}} = mg = 10.9 \times 9.81 \, \text{m/s}^2 \approx 106.8 \, \text{N}$
• Distance traveled upward, $d = 0.37 \, \text{m}$
• Initial speed, $v_0 = 0 \, \text{m/s}$ (we'll assume it starts from rest)

Steps to Solution

1. Calculate the Net Force Acting on the Rocket (Without the Spring):

   The net force acting on the rocket is the difference between the thrust and the gravitational force: $F_{\text{net}} = F_{\text{thrust}} - F_{\text{gravity}} = 210 \, \text{N} - 106.8 \, \text{N} = 103.2 \, \text{N}$

2. Work Done by the Net Force:

   The work done by the net force over the distance $d$ is: $W = F_{\text{net}} \times d = 103.2 \, \text{N} \times 0.37 \, \text{m} = 38.184 \, \text{J}$

3. Use the Work-Energy Theorem:

   According to the work-energy theorem, the work done on the rocket is equal to the change in its kinetic energy: $W = \Delta KE = \frac{1}{2}mv^2 - \frac{1}{2}mv_0^2$ Since the initial speed $v_0 = 0$, the equation simplifies to: $W = \frac{1}{2}mv^2$

4. Solve for $v$:

   Substitute the values and solve for $v$: $38.184 = \frac{1}{2} \times 10.9 \times v^2$ $v^2 = \frac{38.184 \times 2}{10.9} = \frac{76.368}{10.9} \approx 7.004$ $v \approx \sqrt{7.004} \approx 2.65 \, \text{m/s}$

Final Answer

If the rocket weren't attached to the spring, its speed after traveling 37.0 cm would be approximately: $\boxed{2.65 \, \text{m/s}}$

Would you like further details, or do you have any questions?

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Here are five related questions to consider:

1. How would the speed change if the thrust force was increased?
2. What would happen to the rocket's motion if the mass was reduced while keeping the thrust the same?
3. How would you find the speed if the rocket were moving horizontally instead of vertically?
4. How does adding or removing external forces, like the spring, affect the work-energy calculations?
5. Can you calculate the potential energy stored in the spring initially compressed by 22.3 cm?

Tip: When analyzing forces, always separate gravitational forces and other external forces to understand the net effect on motion more clearly.