Question

A spacecraft is coasting toward Mars. The mass of Mars is 6.00×1023kg, and its radius is 3.40×106m. When the spacecraft is 6.92×106m from the center of Mars, the spaceraft's speed is 2910m/s. Later, when the spacecraft is 4.09×106m from the center of Mars, what is its speed?

Answer 1

Answer:

4058.48594 m/s

Explanation:

M = Mass of Mars = $6\times 10^{23}\ kg$

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

$v_i$ = Initial velocity = 2910 m/s

$v_f$ = Final velocity

m = Mass of satellite

$d_i$ = Initial distance of satellite = $6.92\times 10^6\ m$

$d_f$ = Final distance of satellite = $4.09\times 10^6\ m$

The kinetic and potential energy of the system is conserved

$P_i+K_i=P_f+K_f\\\Rightarrow -\frac{GMm}{d_i}+\frac{1}{2}mv_i^2=-\frac{GMm}{d_f}+\frac{1}{2}mv_f^2\\\Rightarrow -\frac{GM}{d_i}+\frac{1}{2}v_i^2=-\frac{GM}{d_f}+\frac{1}{2}v_f^2\\\Rightarrow v_f=\sqrt{2\left(-\frac{GM}{d_i}+\frac{1}{2}v_i^2+\frac{GM}{d_f}\right)}\\\Rightarrow v_f=\sqrt{2\left(-\frac{6.67\times 10^{-11}\times 6\times 10^{23}}{6.92\times 10^6}+\frac{1}{2}2910^2+\frac{6.67\times 10^{-11}\times6\times 10^{23}}{4.09\times 10^6}\right)}\\\Rightarrow v_f=4058.48594\ m/s$

Final velocity is 4058.48594 m/s