A 10.5 kg weather rocket generates a thrust of 240 N . The rocket, pointing upward, is clamped to the top of a vertical spring. The bottom of the spring, whose spring constant is 480 N/m , is anchored to the ground.
a)Initially, before the engine is ignited, the rocket sits at rest on top of the spring. How much is the spring compressed?
b)After the engine is ignited, what is the rocket's speed when the spring has stretched 40.0 cm?
c)For comparison, what would be the rocket's speed after traveling this distance if it weren't attached to the spring?
To solve this problem, we will use the concepts of Newton's laws of motion and the equations of motion.
a) Initially, when the rocket sits at rest on top of the spring, the forces acting on the rocket are the weight (mg) and the force due to the compression of the spring (kx).
According to Newton's second law, the net force acting on the rocket is equal to the mass of the rocket multiplied by its acceleration (F_net = m * a).
In this case, the rocket is at rest, so the acceleration is zero (a = 0). Therefore, we have:
F_net = m * a
F_net = mg + kx
Since the rocket is at rest, the net force acting on it is zero (F_net = 0). Rearranging the equation, we can solve for x, the compression of the spring:
mg + kx = 0
x = -(mg) / k
Now we can substitute the given values:
m = 10.5 kg
g = 9.8 m/s^2
k = 480 N/m
x = -(10.5 kg * 9.8 m/s^2) / 480 N/m
x ≈ -0.214 m
The spring is compressed by approximately 0.214 meters.
b) After the engine is ignited, the rocket will experience an upward force due to the thrust generated by the rocket engine (240 N). The net force acting on the rocket will be the sum of the thrust force and the force due to the spring (F_net = F_thrust + F_spring).
According to Newton's second law, the net force is equal to the mass of the rocket multiplied by its acceleration (F_net = m * a). Rearranging the equation, we can solve for the acceleration:
F_net = m * a
240 N + kx = m * a
a = (240 N + kx) / m
Since the rocket starts from rest, its initial velocity (v0) is zero. The displacement (d) of the rocket when the spring has stretched is 40 cm, which is equivalent to 0.4 m. Using the equation of motion:
d = v0 * t + (0.5) * a * t^2
Substituting the given values:
d = 0.4 m
v0 = 0 m/s
t = unknown
a = (240 N + kx) / m
We need to find t. Rearranging the equation, we get a quadratic equation:
0.5 * a * t^2 + v0 * t - d = 0
Using the quadratic formula, we can solve for t:
t = (-v0 ± sqrt(v0^2 - 4(0.5 * a * (-d)))) / (2 * 0.5 * a)
Substituting the given values:
v0 = 0 m/s
a = (240 N + kx) / m
d = 0.4 m
Calculating t using the quadratic formula will give us the time it takes for the rocket to stretch the spring by 40 cm. Once we know the time, we can determine the rocket's final velocity using the equation:
v = v0 + a * t
c) To find the rocket's speed after traveling 40 cm if it weren't attached to the spring, we can calculate the speed using the equation:
v = sqrt(2 * a * d)
Where a is the rocket's acceleration, which is given by F_thrust / m, and d is the displacement of 40 cm converted to meters (0.4 m). By substituting these values into the equation, we can find the rocket's speed.