Question

Astronomers detect a distant meteoroid moving
along a straight line that, if extended, would pass
at a distance 3RE from the center of the Earth,
where Re is the Earth's radius. The minimum
speed must the meteoroid have if it is not to
collide with the Earth is:​

Answer 1

The minimum speed of the meteroid so that it does not collide with the earth is $\boxed{\frac{1}{2}\sqrt{\frac{GM}{R_e}}}$

Explanation:

If the meteroid is not to collide

Then it will just pass through touching the surface of the earth theoretically

Let the mass of the meteroid be m

Assuming the initial potential energy of meteroid at infinity as zero and its initial velocity as $v_i$, if the final velocity is $v_f$

Then

$U_i+K_i=U_f+K_f$

$\implies 0+\frac{1}{2}mv_i^2=-\frac{GMm}{R_e}+\frac{1}{2}mv_f^2$    (where M is the mass of earth)

if there was no gravitational force acting on the meteroid

then it will have continued on its linear path and would have passed the earth at distance $3R_e$

Thus, if take initially the meteroid at distance  $3R_e$ then finally it is at distance is $R_e$ and there is no external torque is acting only gravitational force in linear direction

Thus, its angular momentum about the centre of the earth will be conserved

Therefore,

$v_i\times 3R_e=v_f\times R_e$

$\implies v_f=3v_i$

Thus,

$v_i^2=-\frac{2GM}{R_e}+(3v_i)^2$

$\implies v_i^2=-\frac{2GM}{R_e}+9v_i^2$

$\implies 8v_i^2=\frac{2GM}{R_e}$

$\implies v_i=\frac{1}{2}\sqrt{\frac{GM}{R_e}}$

Hope this answer is helpful.

Know More:

Q: The mass of the earth is 6 x 1024 kg and that of the moon is 74 x 10 kg. If the  distance between the earth and the moon be 3.84 x 105 km, calculate the force exerted by the earth on the  moon. (G = 6.7 * 10-11 Nm kg)?

Click Here: https://brainly.in/question/16233703

Q: A meteorite burns in the atmosphere before it reaches the earth's surface.what happens to its momentum?

Click Here: https://brainly.in/question/1448270