What is the elastic constant, k, for each band: A cart is pushed into a red elas

Question

What is the elastic constant, k, for each band: A cart is pushed into a red elastic band, the graph (x^2 vs KE of cart) has a slope of .036. The same cart is pushed into a blue and the slope of the line is .023. k represents the spring constant of the spring. The compression or extension ofthe spring is x in this equation.

So far I have: KE=1/2mv2   PE = 1/2kx2   KE+PE = Constant

PE = -m(x2) = 1/2kx2   slope=2/k kblue= 87 J/m2 kred= 56 J/m2 *** Please confirm if this is correct.

Part 2: If you now rotate the apparatus holding your blue elastic band such that it lies horizontally instead of vertically you could now use it to launch objectsvertically.Suppose that you now have a small 200g toy rocket and you use it to push down on the elastic band and stretch it 12 cm.   Determine how high above its launch positionit will travel when you release it.

Explanation / Answer

Part 1) KE + PE = constant

=> KE + kx2/2 = c

=> x2 = (2/k)KE + 2c/k

Slope, m = 2/k

For red band, 2/kred = 0.036

=> kred = 2/0.036 = 55.6 J/m2

Similarly, kblue = 2/0.023 = 90.0 J/m2

Part 2) Elastic potential energy of the cart is converted to the KE of the toy rocket.

So, KE = 0.2v2/2 = 90 * 0.122 / 2

=> v = 0.12 * (90 / 0.2)1/2 = 2.55 m/s

Height attained above the launch point, h = 0.12 + v2/2g = 0.12 + [2.552 / (2 * 9.81)] = 0.45 m

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