Question

A satellite is to be put into an orbit $600\text{km}$ above the surface of the earth. If its vertical velocity after launching is $2400\text{m/s}$ at this height, the magnitude and direction of the impulse required to put the satellite directly into orbit is
$(\text{The mass of the satellite is}60\text{kg}\text{and the radius of the earth is}6400\text{km and g}=10{\text{ms}}^{-2})$.

• A. Impulse = $4.77\times {10}^5$ m/s at an angle ${\tan }^{-1}\left(0.317\right)$ from horizontal (downward).
• B. Impulse = $2.57\times {10}^3$ m/s at an angle ${\tan }^{-1}\left(0.217\right)$ from horizontal (downward).
• C. Impulse = $3.75\times {10}^4$ m/s at an angle ${\tan }^{-1}\left(0.398\right)$ from horizontal (downward).
• D. Impulse = $9.76\times {10}^6$ m/s at an angle ${\tan }^{-1}\left(0.317\right)$ from horizontal (downward).

Answer 1

The correct option is A Impulse = $4.77\times {10}^5$ m/s at an angle ${\tan }^{-1}\left(0.317\right)$ from horizontal (downward).

$\text{Initial velocity}=2400\text{m/s}\text{vertical}$
$\text{Final velocity}=v\text{m/s horizontal}$
$\text{Centrifugal force}=\text{Gravitational force}$
$\Rightarrow \frac{m{v}^2}{R}=\frac{GMm}{R^2}$
where R= distance from centre, M= mass of earth = $5.97\times {10}^{24}kg$, m=mass of satellite
$\Rightarrow v=\sqrt{\frac{GM}{R}}$

$\Rightarrow v=\sqrt{\frac{(6.673\times {10}^{-11})(5.97\times {10}^{24})}{(6400+600)\times 1000}}$
$\Rightarrow v=7543.948\text{m/s}$
Thus impulse required in horizontal direction$=mv=60\times 7543.948=4.526\times {10}^5\text{kgm/s}$
Impulse is also required in vertically downward direction to reduce the vertical velocity to 0.
Impulse required in vertically downward direction $=60\times 2400=1.44\times {10}^{5}kgm/s$
Thus
Net impulse $=\sqrt{{1.44}^2+{4.526}^2}\times {10}^5=4.74\times {10}^5\cong \mathrm{4..77}\times {10}^{5}kgm/s$
Angle impulse makes with horizontal (in downward direction because vertical component of impulse is in downward direction)
$=ta{n}^{-1}\frac{1.44}{4.526}=ta{n}^{-1}0.317$ downwards from horizontal
Thus correct option is (a)