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A certain volcano on earth can eject rocks vertically to a maximum height H. (a) How high (in terms of H) would these rocks go if a volcano on Mars ejected them with the same initial velocity? The acceleration due to gravity on Mars is 3.71 m/s2; ignore air resistance on both planets. (b) If the rocks are in the air for a time T on earth, for how long (in terms of T) would they be in the air on Mars?

Answer :

Manetho

Explanation:

a) By conservation of energy we can write

mgh on earth = mgh on mars.

[tex]mg_Eh_E=mg_Mh_M[/tex]

M,E are earth and mars respectively.

[tex]h_m =\frac{g_E}{g_M}\times h_E[/tex]

[tex]h_m=\frac{9.81}{3.71}\times h_E[/tex]

h_m= 2.64 h_E

b) Consider the time taken for the rock to reach the top of its trajectory. By symmetry, this is T/2. Inserting this into the kinematics equation v = u+at, we get the following two sets of equations:

final velocities will be zero v= 0

[tex]0= v- g_E\frac{T}{2}[/tex]

[tex]0=v- g_M\frac{T_M}{2}[/tex]

This gives [tex]2v= g_ET_E=g_mT_m[/tex] and

therefore,

[tex]T_M= 2.64T_E[/tex]

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