Question

An instrument package is projected vertically upward to collect data near the top of the Earth's atmosphere (at an altitude of about 840 km ).

Part A

What initial speed is required at the Earth's surface for the package to reach this height?

Express your answer using two significant figures.

Part B

What percentage of the escape speed is this initial speed?

Express your answer using two significant figures.

Answer 1

part A

at earth surface, total energy = P.E. + K.E.

= -GMm/R + mv_i²/2

at the top of earth’s atmosphere, total energy = P.E. + K.E.

= -GMm/(R+h) + mv_f²/2

as per energy conservation,

-GMm/R + mv_i²/2 = -GMm/(R+h) + mv_f²/2

As per given conditions

-GMm/R + mv_i²/2 = -GMm/(R+h) + 0 (as v_f =0)

-GM/R + v_i²/2 = -GM/(R+h)

v_i² = 2GM{1/R - 1/(R+h)} = 79.666x10¹³(1.813x10^-8) = 144.434x10⁵

v_i = 3800.447 m/s

and as per required format, the answer is v_i = 3800 m/s.

part B

as escape speed = 11200 m/s.

hence the above mentioned speed is (3800/11200)(100)=33.93 % of escape speed.

and as per required format, the answer 34%.