Question

The 2004 landings of the Mars rovers Spirit and Opportunity involved many stages, resulting in each probe having zero vertical velocity about 12 m above the surface of Mars. Determine
(a) the time required for the final free-fall descent of the probes
(b) the vertical velocity at impact. The mass of Mars is 6 . 419 × 10 23 kg , and its radius is 3 . 397 × 10 6 m

Answer 1

Answer:

2.54334 seconds

9.4364 m/s

Explanation:

t = Time taken

u = Initial velocity

v = Final velocity

s = Displacement

a = Acceleration

g = Acceleration due to gravity

G = Gravitational constant = 6.67 × 10⁻¹¹ m³/kgs²

M = Mass of Mars = $6.419\times 10^{23}\ kg$

R = Radius of Mars = $3.397\times 10^6\ m$

The acceleration due to gravity of a planet is given by

$g=\frac{GM}{R^2}\\\Rightarrow g=\frac{6.67\times 10^{-11}\times 6.419\times 10^{23}}{(3.397\times 10^6)^2}\\\Rightarrow g=3.71024\ m/s^2$

Equation of motion

$s=ut+\frac{1}{2}at^2\\\Rightarrow 12=0t+\frac{1}{2}\times 3.71024\times t^2\\\Rightarrow t=\sqrt{\frac{12\times 2}{3.71024}}\\\Rightarrow t=2.54334\ s$

The time taken to fall 12 m in Mars is 2.54334 seconds

$v=u+at\\\Rightarrow v=0+3.71024\times 2.54334\\\Rightarrow v=9.4364\ m/s$

The vertical velocity at impact is 9.4364 m/s