Aime F. answered • 12/18/22
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PhD in Physics (Yale), have taught Methods of Engineering Analysis
The potential energy u of the spring compressed a distance d = 0.046 m is u = kd²/2. The kinetic energy e imparted to the mass m = 0.418 kg is e = mv²/2. By conservation of energy, u = e provides one equation in the 2 unknowns k and v. I believe more information need be provided for the answers to be calculated.
Aime F.
You did not include "the figure below" but if we knew the height h of the track above the ground then we can further equate kd²/2 = mv²/2 = mV²/2 – mgh for the speed V when it hits the ground. We still have 2 equations in 3 unknowns k, v and V, insufficient information.
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12/18/22
Aime F.
If we knew how far X the mass travelled horizontally from the track end then we could eliminate its falling time T between X = vT and h = gT²/2 to get h = gX²/2v², a 3rd equation whence we would have 3 equations for the 3 unknowns k, v, V. Please check that you include all information in your questions.
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12/18/22
Aime F.
If we solve the differential eq. m x''(t) = –k x(t), x(0) = –d, x'(0) = 0 we get x(t) = –d cos(√(k/m)t) for 0 < t < π/2√(k/m) = t₀. Therefore v = d √(k/m) sin(√(k/m)t₀) = d √(k/m) but that's the same as you got from u = e above.12/18/22