An obnoxious brat assigns his 0.0221-kilogram pet mouse Mickey the role of stunt mouse by stuffing him into a spring gun and shooting him up into the air at some angle. During the launch process the spring transfers 0.707 J of energy into the \'mouse Earth gravitational fields\' system. When Mickey reaches his maximum height, he has a horizontal velocity with a speed of 2.37 m/s. How high above his initial location in the gun does Mickey soar? (In case you are worried, a miniature parachute opens at this point and Mickey lands safely. The brat is slightly less obnoxious than we thought.) Take g = 9.81 m/s2.
can somebody work it out to show me not just give the formulas, please , thanks
Sure! To find the height above the initial location where Mickey soars, we can use the principle of conservation of mechanical energy. The total mechanical energy of the system remains constant throughout the motion.
The total mechanical energy (E) of the system includes both the kinetic energy (K) and the gravitational potential energy (PE). Mathematically, it can be written as:
E = K + PE
Initially, Mickey is at rest and has no potential energy. Hence, the initial total mechanical energy (Ei) is equal to the initial kinetic energy (Ki).
Ei = Ki = 0.5 * m * v_initial^2 ------------ (1)
At the maximum height, Mickey momentarily comes to rest. Therefore, at this point, there is no kinetic energy. The total mechanical energy (Emax) is equal to the gravitational potential energy (PEmax) at maximum height.
Emax = PEmax = m * g * h_max ------------ (2)
The energy transferred by the spring gun (0.707 J) is equal to the change in mechanical energy (∆E) between the initial and maximum height.
∆E = Emax - Ei = m * g * h_max - 0.5 * m * v_initial^2 ------------ (3)
Now we can substitute the given values into the equations:
m = 0.0221 kg (mass of Mickey)
g = 9.81 m/s^2 (acceleration due to gravity)
v_initial = 2.37 m/s (initial horizontal velocity)
Substituting these values into equation (1), we can calculate the initial mechanical energy (Ei):
Ei = 0.5 * 0.0221 kg * (2.37 m/s)^2 = 0.115 J
Using the value of Ei and the energy transferred (∆E = 0.707 J) in equation (3), we can find the maximum height (h_max):
0.707 J = m * g * h_max - 0.5 * m * (2.37 m/s)^2
0.707 J = 0.0221 kg * 9.81 m/s^2 * h_max - 0.5 * 0.0221 kg * (2.37 m/s)^2
Solving for h_max, we have:
h_max = (0.707 J + 0.5 * 0.0221 kg * (2.37 m/s)^2) / (0.0221 kg * 9.81 m/s^2)
h_max ≈ 0.0818 meters
Therefore, Mickey soars approximately 0.0818 meters (or 8.18 cm) above his initial location in the gun before his parachute opens, and he lands safely.
Note: In this calculation, we assumed no energy losses due to air resistance.