Am I lactose intollerant?

Prince Lactose the Intollerant and the spring-launched satellite?

• Prince Lactose the Intollerant wanted to be the first to launch a satellite. He placed a 2.0 kg payload on top of a very stiff 2.0-m-long spring with a spring constant of 50,000N/m. The the prince had his strongest men use a winch to crank the spring down to a length of 80 cm. When released, the spring shot the payload straight up. How high did it go?

• Answer:

  The PE(h) of the compressed spring and the satellite = (kdelX^2)/2 + mgh; where k = 50K N/m, m = 2 kg, and h = 2 - .8 = 1.2 m, the height of the satellite above the ground while resting on the compressed spring. From the conservation of energy total energy TE(h = 1.2) = PE(h) = (kdelX^2)/2 + mg(1.2). Upon decompression TE(h = 2) = 1/2 mV^2 + mg2 = KE(2) + PE(2); where KE(2) is the kinetic energy of the satellite as it leaves the spring at h = 2 m above ground. Finally, TE(H) = mgH; where H is the maximum height and it includes the PE's for the 1.2 or 2 meter heights; so H is height above ground. From the conservation of energy we have TE(h = 1.2) = (kdelX^2)/2 + mg(1.2) = mgH = TE(H); so H = ((kdelX^2)/2 + mg(1.2))/mg = ((50000*(.80)^2)/2 + 2*9.81*1.2)/(2*9.81); g = 9.81 m/sec^2 in MKS units, you have the numbers you can do the math and get an answer in meters. [Hint: Prince Lactose did not win any altitude awards for his milktoast launch.] Conservation of energy is the physics. But some things you need to be aware of. First, when the spring is compressed it has potential energy to be sure. But so does the satellite by virtue of being 1.2 meters above the ground while sitting on the compressed spring. Thus the H value includes the initial height; so H is the height above ground. It also includes the 2 meter launch height and, again, H is above ground. Note: The first answer is incorrect because it failed to account for the fact that the satellite itself had a potential energy while sitting on the compressed spring. And that potential before launch contributes to the potential energy at h = H max height.