A 11.6 kg weather rocket generates a thrust of 215.0 N. The rocket, pointing upward, is clamped to the top of a vertical spring. The bottom of the spring, whose spring constant is 399.0 N/m, is anchored to the ground. Initially, before the engine is ignited, the rocket sits at rest on top of the spring.
a)How much is the spring compressed?
b)After the engine is ignited, what is the rocket's speed when the spring has stretched 16.1 cm past its natural length?
c)What would be the rockets speed after travelling the distance if it weren't tied down to the spring?
a)
f=-kx
f=>11.6*(-9.8)=-399.0x
solve for x
x=0.285m
To answer these questions, we'll need to use some principles from physics, specifically Hooke's Law and Newton's laws of motion.
a) To find how much the spring is compressed, we can use Hooke's Law, which states that the force exerted by a spring is directly proportional to the displacement of the spring from its equilibrium position. Mathematically, this can be stated as F = -kx, where F is the force applied by the spring, k is the spring constant, and x is the displacement of the spring. In this case, the force applied by the spring is equal to the weight of the rocket, which can be calculated as mg, where m is the mass and g is the acceleration due to gravity. So we have mg = -kx. Rearranging this equation, we get x = -mg/k.
Plugging in the values:
m = 11.6 kg
g = 9.8 m/s^2 (approximate value for gravity on Earth)
k = 399.0 N/m
x = -(11.6 kg * 9.8 m/s^2) / (399.0 N/m)
Calculating this gives us the amount of compression of the spring.
b) To find the rocket's speed when the spring has stretched 16.1 cm past its natural length, we'll need to consider the forces acting on the rocket. Initially, before the engine is ignited, the only force acting on the rocket is the force exerted by the compressed spring, which is given by Hooke's Law as F = -kx. However, once the engine is ignited, the thrust force generated by the rocket also comes into play. According to Newton's second law of motion, the net force acting on an object is equal to its mass multiplied by its acceleration. In this case, the mass refers to the total mass of the rocket.
So we have the following equation: -(kx) + thrust = mass * acceleration
We can rearrange this equation to solve for acceleration: acceleration = (thrust - kx) / mass
To find the speed, we can integrate the acceleration function with respect to time to get the velocity function, and then solve for the velocity when the spring has stretched 16.1 cm past its natural length.
c) To find the rocket's speed after traveling the distance if it weren't tied down to the spring, we'll need to use the principles of work and energy. The work done on an object is equal to the change in its kinetic energy. The initial potential energy stored in the spring will be converted into kinetic energy as the rocket gets free from the spring's clamps.
The potential energy stored in the compressed spring can be calculated as (1/2)kx^2, and this will be equal to the kinetic energy of the rocket after it has traveled the given distance. By using the equation for kinetic energy, (1/2)mv^2, we can set these two energies equal to each other and solve for the velocity v.