Figure 13-53 is a graph of the kinetic energy Kof an asteroid versus its distance rfrom Earth’s center, as the asteroid falls directly in toward that center. (a) What is the (approximate) mass of the asteroid? (b) What is its speed atr=1.945×107m ?

Short Answer

Expert verified

Answer:

  1. The mass of asteroid ism=1.0×103kg
  2. The speed of asteroid is at r=1.945×107misv=1.5×103m/s

Step by step solution

01

Significance of Newton’s law of universal gravitation

Every particle in the universe is attracted to every other particle with a force that is directly proportional to the product of the masses and inversely proportional to the square of the distance, according to Newton's Law of Universal Gravitation.

We can use the concept of energy conservation law. When the asteroid falls towards the earth, its gravitational potential energy is converted into kinetic energy.

Formula:

U=-GMmrK=12mv2

Where.

G is the gravitational constant ( 6.67×10-11m3/kg·s2)

M is the mass earth

K is the kinetic energy

U is the potential energy

v is the speed of object

m is the mass of asteroid

02

(a) Determining the mass of asteroid

According to the energy conservation law,

K1-U1=K2-U2K1-GMmr1=K2-GMmr2GMmr2-GMmr1=K2-K1m=K2-K1GM1r2-1r1=3.2×109J-2.2×109J6.67×10-11N.m2kg25.98×1024kg11.77×107m-11.85×107m=1.0×103kg

03

(b) Determining the speed of asteroid

At a distance r=1.945×107m, the kinetic energy of the asteroid is K=1.2×109J. Hence the speed of the asteroid is

K=12mv2

dv=2Km=2×1.2×109J1.0×103kg=1.5×103m/s

The speed of asteroid is at r = 1.945 x 107 is v =1.5×103m/s

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