Answer:
The maximum height reached by the ride after it was pulled to the ground is 51.6 m.
The given parameters;
- Distance between the two towers, d = 31 m
- Height of the tower, h = 76 m
- Unstretched length of the band, L₀ = 41 m
- Elastic constant of the band, k = 310 N/m
The distance half-way between the bands;
[tex]\frac{d}{2} = \frac{31}{2} = 15.5 \ m[/tex]
The maximum length of the band when stretched is calculated as;
[tex]c^2 = 15.5^2 + 76^2\\\\c^2 = 6016.25\\\\c = \sqrt{6016.25} \\\\c = 77.57 \ m[/tex]
The extension of the elastic band;
x = 77.57 m - 41 m
x = 36.37 m
The elastic potential energy stored in the band;
[tex]E = \frac{1}{2} kx^2\\\\E = \frac{1}{2} \times 310 \times (36.57)^2\\\\E = 207,291.56 \ J[/tex]
The elastic potential energy of the elastic band will be converted into kinetic energy of the ride and the speed of the ride is calculated as;
[tex]E = \frac{1}{2} mv^2\\\\207,291.56 = \frac{1}{2} \times 410 \times v^2\\\\v^2 = \frac{207,291.56}{(0.5\times 410)} \\\\v^2 = 1011.178\\\\v = \sqrt{1011.178} \\\\v = 31.8 \ m/s[/tex]
The maximum height reached by the ride is calculated as;
[tex]P.E = K.E\\\\mgh = \frac{1}{2} mv^2\\\\h = \frac{v^2}{2g} \\\\h = \frac{(31.8)^2}{(2\times 9.8)} \\\\h = 51.6 \ m[/tex]
Thus, the maximum height reached by the ride is 51.6 m.
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