A 10.2 kg weather rocket generates a thrust of 200N. The rocket, pointing upward, is clamped to the top of a vertical spring. The bottom of the spring, whose spring constant is 500N/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 being compressed?
ANSWER. 0.2m or 20 cm
b) After the engine is ignited, what is the rocket's speed when the spring has stretched 40 cm? For comparison, what would the rocket's speed be after traveling this distance if it weren't attached to the spring?
Help with B please!!!
B: After the engine is ignited, the rocket's speed when the spring has stretched 40 cm can be determined using conservation of mechanical energy.
First, let's find the potential energy stored in the spring when it is stretched by 40 cm. The potential energy in a spring is given by the equation:
Potential energy = (1/2)kx^2
where k is the spring constant and x is the displacement of the spring from its equilibrium position.
Given:
Spring constant, k = 500 N/m
Displacement, x = 40 cm = 0.4 m
Potential energy = (1/2) * 500 N/m * (0.4 m)^2
= 40 J
Since the spring is clamped to the rocket, this potential energy will be converted into kinetic energy of the rocket. Therefore, equating potential energy to kinetic energy:
Kinetic energy = Potential energy
(1/2)mv^2 = 40 J
where m is the mass of the rocket and v is its velocity.
Given:
Mass of the rocket, m = 10.2 kg
Simplifying the equation:
(1/2) * 10.2 kg * v^2 = 40 J
v^2 = (40 J / (1/2) * 10.2 kg)
v^2 = 7.8431 m^2/s^2
Taking the square root of both sides:
v ≈ √(7.8431 m^2/s^2)
v ≈ 2.8 m/s
Therefore, the rocket's speed when the spring has stretched 40 cm is approximately 2.8 m/s.
For comparison, if the rocket were not attached to the spring, its speed after traveling the same distance would depend on the net force acting on it. Since the thrust force generated by the rocket engine is 200 N, the rocket's acceleration without the spring would be given by Newton's second law:
Net force = mass * acceleration
200 N = 10.2 kg * acceleration
acceleration = 200 N / 10.2 kg
acceleration ≈ 19.61 m/s^2
Using the kinematic equation:
v^2 = u^2 + 2as
where u is the initial velocity (zero in this case), a is the acceleration, and s is the displacement of 0.4 m, we can solve for v:
v^2 = 0 + 2 * 19.61 m/s^2 * 0.4 m
v^2 ≈ 15.69 m^2/s^2
Taking the square root:
v ≈ √(15.69 m^2/s^2)
v ≈ 3.96 m/s
Therefore, the rocket's speed after traveling 40 cm without the spring would be approximately 3.96 m/s.
To find the rocket's speed when the spring has stretched 40 cm, we can use the principle of conservation of energy.
The initial potential energy stored in the spring when it is compressed by a certain distance is given by:
Potential energy (U) = 0.5 * k * x^2
Where:
k is the spring constant, which is 500 N/m
x is the compression or stretch of the spring
a) Initially, before the engine is ignited, the rocket sits at rest on top of the spring. So the spring is compressed. We need to find the compression of the spring.
Given:
Mass of the rocket (m) = 10.2 kg
Gravitational acceleration (g) = 9.8 m/s^2
Thrust (F) = 200 N
Spring constant (k) = 500 N/m
To find the compression, we can equate the force due to the compression of the spring (Fspring) to the force due to gravity acting on the rocket (mg):
Fspring = mg
From this equation, we can solve for x (compression):
kx = mg
x = mg/k
Let's calculate the compression:
x = (10.2 kg * 9.8 m/s^2) / (500 N/m)
x ≈ 0.2 m or 20 cm
Therefore, initially before the engine is ignited, the spring is compressed by 0.2 m or 20 cm.
b) After the engine is ignited, when the spring has stretched by 40 cm, we need to find the rocket's speed.
To find the rocket's speed, we can again use the conservation of mechanical energy:
Initial energy = Final energy
The initial energy includes the potential energy stored in the compressed spring and the kinetic energy due to the rocket's motion. The final energy includes the potential energy stored in the stretched spring and the kinetic energy due to the rocket's motion.
The initial potential energy is given by:
Potential energy (initial) = 0.5 * k * x^2
Where x is the initial compression of the spring, which is 20 cm or 0.2 m.
The final potential energy is given by:
Potential energy (final) = 0.5 * k * x^2
Where x is the final stretch of the spring, which is 40 cm or 0.4 m.
The initial kinetic energy is zero since the rocket is initially at rest.
The final kinetic energy is given by:
Kinetic energy (final) = 0.5 * m * v^2
Where m is the mass of the rocket, which is 10.2 kg, and v is the velocity of the rocket.
Setting up the equation:
0.5 * k * x^2 + 0.5 * m * 0^2 = 0.5 * k * x^2 + 0.5 * m * v^2
Simplifying:
0 + 0 = 0.5 * k * x^2 + 0.5 * m * v^2
Since the rocket is initially at rest, the equation becomes:
0 = 0.5 * k * x^2 + 0.5 * m * v^2
Solving for v:
0.5 * m * v^2 = -0.5 * k * x^2
v^2 = (-0.5 * k * x^2) / m
v = sqrt((-0.5 * k * x^2) / m)
Let's calculate the rocket's speed:
v = sqrt((-0.5 * 500 N/m * (0.4 m)^2) / 10.2 kg)
v ≈ 2.44 m/s
Therefore, when the spring has stretched by 40 cm, the rocket's speed is approximately 2.44 m/s.
For comparison, if the rocket weren't attached to the spring, it would continue to accelerate due to the thrust generated by the engine. Assuming no other forces are acting on the rocket, we can use the kinematic equation:
v = u + at
Since the initial velocity (u) is zero, and the rocket experiences constant acceleration due to the thrust (F = ma), the equation becomes:
v = at
Using the given data:
Thrust (F) = 200 N
Mass of the rocket (m) = 10.2 kg
v = (F/m) * t
Substituting the values:
v = (200 N / 10.2 kg) * (0.4 m / (200 N/10.2 kg))
v ≈ 0.78 m/s
Therefore, if the rocket weren't attached to the spring, its speed after traveling 40 cm would be approximately 0.78 m/s.
To find the rocket's speed when the spring has stretched 40 cm, we can use conservation of mechanical energy. The initial potential energy stored in the compressed spring is equal to the work done by the rocket's thrust plus the final kinetic energy of the rocket.
Let's break down the steps to get the solution:
Step 1: Calculate the initial potential energy of the compressed spring.
The potential energy of a spring is given by the equation: PE = (1/2)kx^2, where k is the spring constant and x is the displacement of the spring from its equilibrium position.
In this case, the spring is compressed by 0.2 m (as found in part a) and the spring constant is 500 N/m.
Initial potential energy (PE) = (1/2) * 500 * (0.2)^2 = 10 J (Joules)
Step 2: Find the work done by the rocket's thrust.
The work done by a force is given by the equation: Work = force * distance * cos(theta), where theta is the angle between the force and displacement vectors. Since the force and displacement vectors are aligned in this case, theta is 0 degrees.
The force applied by the rocket's thrust is 200 N (given).
The distance over which the force is applied is the displacement of the spring, which is 0.4 m (given).
Work done by the rocket's thrust = 200 * 0.4 * cos(0) = 80 J
Step 3: Calculate the final kinetic energy of the rocket.
The final kinetic energy of the rocket is equal to the initial potential energy of the spring plus the work done by the rocket's thrust.
Final kinetic energy = Initial potential energy + Work done by thrust
Final kinetic energy = 10 J + 80 J = 90 J
Step 4: Find the rocket's speed.
The kinetic energy of an object is given by the equation: KE = (1/2)mv^2, where m is the mass of the object and v is the speed.
Rearranging the equation, we have: v = sqrt(2KE/m)
The mass of the rocket is 10.2 kg (given).
The final kinetic energy is 90 J (calculated above).
Rocket's speed = sqrt(2 * 90 / 10.2) = sqrt(1.7647) = 1.33 m/s (rounded to 2 decimal places)
To compare this with the rocket's speed if it weren't attached to the spring, you would need to have information about the additional forces acting on the rocket and any other energy changes during the motion.
Why would anyone attach a rocket to a spring?
I suggest using conservation of energy for this problem. Power delivered to Rocket = (Thrust)*(distance travelled, X)
After you subtract the spring potential energy (1/2)kX^2 , the remainder is the kinetic energy of the rocket.