So if I have a rocket going to a certain height - I know its mass, and I know its total impulse, and the height it reaches -how do I calculate for another height? For example, 50Ns is total impulse, mass is 550g, and height is 750m. How do I calculate the impulse needed to reach 850m? This problem isn't in my homework, so I'm not interested in the solution, just its pathway.

To calculate the impulse needed to reach a different height, you can use the principle of conservation of energy. Here's the step-by-step pathway to solve the problem:

1. Start by converting the mass to kilograms. In this case, the mass is given as 550 grams, so you need to divide it by 1000 to convert it to kilograms. So the mass is 0.55 kg.

2. Next, let's calculate the initial potential energy (PEi) at the initial height of 750m. The formula for potential energy is PE = mgh, where m is the mass, g is the acceleration due to gravity (approximately 9.8 m/s^2), and h is the height. Plugging in the values, we get PEi = 0.55 kg * 9.8 m/s^2 * 750 m.

3. Now, let's calculate the final potential energy (PEf) at the desired height of 850m using the same formula. So PEf = 0.55 kg * 9.8 m/s^2 * 850 m.

4. Since the rocket is only affected by gravity, the change in potential energy (ΔPE) from the initial height to the final height is given by ΔPE = PEf - PEi.

5. Now, let's express the change in potential energy in terms of impulse. Recall that impulse (J) is the product of force and time, and force is the rate of change of momentum. In this case, the impulse is given as 50 Ns. Impulse is also equal to the change in momentum, which is equal to the change in mass times the change in velocity (J = Δm * Δv).

6. Assuming the mass of the rocket remains constant, we can rewrite the impulse equation as J = m * Δv, where Δv is the change in velocity.

7. Since the height reached by the rocket is directly related to its velocity, we can express the change in velocity as Δv = v2 - v1, where v2 is the final velocity at 850m and v1 is the initial velocity at 750m.

8. Rearranging the equation, we have Δv = v2 - v1 = (850 m - 750 m). Note that we are not given any specific values for the velocities, but we'll keep it general for the explanation.

9. Substituting the equation for Δv into the impulse equation, we get J = m * (v2 - v1).

10. We can now solve for the final velocity (v2) in terms of impulse and the other variables: v2 = (J / m) + v1.

11. Finally, we can substitute the values of the known variables into the equation and calculate the final velocity (v2). However, if you're only interested in finding the impulse needed to reach a different height, you can stop here without substituting any specific values.

By following this pathway, you can determine the impulse needed to reach a different height using the principle of conservation of energy and the equation for impulse.