A 9.50 kg rocket produces a thrust of 365 N. The rocket is pointed upward and the bottom of the rocket is attached to, and is resting on, a spring with a spring constant of 320 N/m. If the spring compressed 0.291 m prior to launch, what is the rocket's speed when the rocket is launched and the spring has stretched 0.378 m?
Assuming the rocket mass has not changed.....???
spring: stored energy released: k(.291^2)-k(.378^2)
energy from engine: 365*(.291+.378) (force*distance)
add those energies, then set equal to 1/2 mv^2, solve for v
To solve this problem, we will use the principle of conservation of energy. We can calculate the rocket's speed by comparing the potential energy stored in the spring to the kinetic energy of the rocket.
First, let's calculate the potential energy stored in the spring when it is compressed by 0.291 m. The formula for potential energy in a spring is given by:
Potential Energy = (1/2) * k * x^2
where k is the spring constant and x is the displacement of the spring. Plugging in the values we have:
Potential Energy = (1/2) * 320 N/m * (0.291 m)^2
Potential Energy = 13.95 J
Next, let's calculate the kinetic energy of the rocket when the spring has stretched 0.378 m. Since the rocket produces a thrust force of 365 N, we can calculate the work done on the rocket by the thrust force using the formula:
Work = Force * Displacement
Work = 365 N * 0.378 m
Work = 138.27 J
By the principle of conservation of energy, the work done on the rocket should be equal to the change in potential energy:
Work = Change in Potential Energy
138.27 J = Potential Energy - 13.95 J
Taking into account that potential energy is negative (as the spring is compressed), the equation becomes:
138.27 J = -13.95 J - Potential Energy
138.27 J + 13.95 J = -Potential Energy
152.22 J = -Potential Energy
Now, let's calculate the rocket's kinetic energy at launch when the spring has stretched 0.378 m. We know that the total mechanical energy (kinetic energy + potential energy) is conserved, so:
Kinetic Energy at Launch = Total Energy - Potential Energy
The total energy can be calculated by adding the potential energy to the work done on the rocket:
Total Energy = Potential Energy + Work
Total Energy = 13.95 J + 138.27 J
Total Energy = 152.22 J
Therefore, the rocket's kinetic energy at launch is:
Kinetic Energy at Launch = 152.22 J - 13.95 J
Kinetic Energy at Launch = 138.27 J
Finally, we can use the formula for kinetic energy to calculate the rocket's speed at launch:
Kinetic Energy = (1/2) * mass * velocity^2
Solving for velocity:
velocity^2 = (2 * kinetic energy) / mass
velocity^2 = (2 * 138.27 J) / 9.50 kg
velocity^2 = 29.16 m^2/s^2
Taking the square root of both sides:
velocity = √(29.16 m^2/s^2)
velocity ≈ 5.40 m/s
Therefore, the rocket's speed when launched and the spring has stretched 0.378 m is approximately 5.40 m/s.