To find how much the spring is compressed initially before the rocket engine is ignited, we can set up the equation for equilibrium.

Since the rocket sits at rest on top of the spring, the force due to gravity on the rocket will compress the spring by an amount that balances this gravitational force. Here are the steps to solve for the spring compression:

1. Identify the forces:

   • The gravitational force acting on the rocket: $F_{\text{gravity}} = mg$
   • The spring force that balances this gravitational force at equilibrium: $F_{\text{spring}} = kx$

   where:

   • $m = 10.9 \ \text{kg}$ (mass of the rocket),
   • $g = 9.8 \ \text{m/s}^2$ (acceleration due to gravity),
   • $k = 480 \ \text{N/m}$ (spring constant),
   • $x$ is the compression of the spring.

2. Set up the equation: At equilibrium, the gravitational force is balanced by the spring force: $F_{\text{gravity}} = F_{\text{spring}}$ $mg = kx$

3. Solve for $x$: Rearrange the equation to find $x$: $x = \frac{mg}{k}$ Substitute the known values: $x = \frac{(10.9 \ \text{kg})(9.8 \ \text{m/s}^2)}{480 \ \text{N/m}}$

4. Calculate: $x = \frac{106.82 \ \text{N}}{480 \ \text{N/m}}$ $x \approx 0.222 \ \text{m}$

So, the spring is compressed by approximately 0.222 meters or 22.2 cm.

Would you like further details on any step or additional assistance with related problems?

---

Here are five related questions to deepen your understanding:

1. What would be the compression of the spring if the mass of the rocket increased to 15 kg?
2. How would the spring compression change if the spring constant was doubled?
3. What is the potential energy stored in the spring at the initial compression?
4. If the rocket were to be unclamped and thrust upwards, how would we calculate the initial acceleration?
5. How would air resistance affect the rocket’s upward motion if it were launched?

Tip: When working with springs, remember Hooke's Law, which states that the force exerted by a spring is proportional to its displacement, defined as $F = kx$, where $k$ is the spring constant and $x$ is the displacement.