Q. A 10.6 kg weather rocket generates a thrust of 226.0 N. The rocket, pointing upward, is clamped to the top of a vertical spring. The bottom of the spring, whose spring constant is 402.0 N/m, is anchored to the ground. Initially, before the engne is ignited, the rocket sits at rest on top of the spring.
A: After the engine is ignited, what is the rocket's speed when the spring has stretched 19.4 cm past its natural length?
B: What would be the rocket's speed after travelling the distance if it weren't tied down to the spring?
Attemp:
i used 1/2kx^2+Fthrust(d) = 1/skx^2 + mg(d) + 1/2mv^2 to solve (A) but i'm getting the wrong answer and i also used 3 other equations to solve this but they all wrong.
i have no idea how to start part (B)
Ok, net force upward pulling on the spring is thrust-mg
That net force * distance= 1/2 k distance^2 + 1/2 m v^2
Assume mass of rocket stays same, in reality, it does not as fuel is burned. Make certain distance is in meters, v in m/s
If not tied to spring..
Vf^2=2ad where a= net force/mass
Thank You!!!
To solve part A of the problem, you correctly identified the equation that needs to be used, which is the conservation of energy equation. However, it seems that there might have been some error in your calculations. Let's break down the steps to find the rocket's speed when the spring has stretched 19.4 cm past its natural length.
Step 1: Determine the potential energy stored in the spring when it is stretched by 19.4 cm.
The potential energy stored in a spring can be calculated using the formula: PE = (1/2)kx^2, where k is the spring constant and x is the displacement from the spring's natural length.
Given:
- Spring constant, k = 402.0 N/m
- Displacement, x = 19.4 cm = 0.194 m
PE = (1/2)(402.0)(0.194)^2
PE = 7.8532 J
Step 2: Determine the work done by the thrust force.
The work done by the thrust force can be calculated by multiplying the force by the displacement in the direction of the force.
Given:
- Thrust force, Fthrust = 226.0 N
- Displacement, d = 0.194 m
Work = Fthrust * d
Work = 226.0 * 0.194
Work = 43.844 J
Step 3: Equate the total energy before and after the engine ignition.
Initial (before engine ignition) energy = Final (after engine ignition) energy
(1/2)kx^2 + Fthrust(d) = (1/2)mv^2
Substituting the values we calculated:
7.8532 + 43.844 = (1/2)(10.6)v^2
51.6972 = 5.3v^2
v^2 = 9.74830188
v = √9.74830188
v = 3.12 m/s
Therefore, the rocket's speed when the spring has stretched 19.4 cm past its natural length is approximately 3.12 m/s.
Now let's move on to part B of the problem.
In part B, we need to determine the rocket's speed after traveling the same distance if it weren't tied down to the spring. Since the rocket is no longer influenced by the spring force, we can use the work-energy theorem to find the speed.
The work done on the rocket is equal to the change in kinetic energy.
Work = ΔKE = 1/2 mv^2
The work done is equal to the force applied multiplied by the displacement. Here, the force applied is the thrust force, and the displacement is the same distance the spring stretched (0.194 m).
Fthrust(d) = 226.0 * 0.194 = 43.844 J
Therefore, the work done by the thrust force is 43.844 J.
Now, equate the work done by the thrust force to the kinetic energy:
43.844 J = 1/2 m v^2
We already know the mass of the rocket is 10.6 kg.
43.844 = 1/2 * 10.6 * v^2
Simplifying,
v^2 = (2 * 43.844) / 10.6
v^2 ≈ 8.25
v ≈ √8.25
v ≈ 2.87 m/s
Thus, the rocket's speed after traveling the same distance, if it weren't tied down to the spring, is approximately 2.87 m/s.