The launch velocity is 4.8 m/s
Explanation:
We can solve this problem by applying the law of conservation of energy. In fact, the mechanical energy of the hopper (equal to the sum of the potential energy + the kinetic energy) is conserved. So we can write:
[tex]U_i +K_i = U_f + K_f[/tex]
where:
[tex]U_i[/tex] is the initial potential energy, at the bottom
[tex]K_i[/tex] is the initial kinetic energy, at the bottom
[tex]U_f[/tex] is the final potential energy, at the top
[tex]K_f[/tex] is the final kinetic energy, at the top
We can rewrite the equation as:
[tex]mgh_i + \frac{1}{2}mu^2 = mgh_f + \frac{1}{2}mv^2[/tex]
where:
m is the mass of the hopper
[tex]g=9.8 m/s^2[/tex] is the acceleration of gravity
[tex]h_i = 0[/tex] is the initial height
u is the launch speed of the hopper
[tex]h_f = 1.20 m[/tex] is the maximum altitude reached by the hopper
v = 0 is the final speed (which is zero when the hopper reaches the maximum height)
Solving the equation for u, we find the launch speed of the hopper:
[tex]u=\sqrt{2gh_g}=\sqrt{2(9.8)(1.20)}=4.8 m/s[/tex]
Learn more about kinetic energy and potential energy:
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