Question

The computational model below shows the total energy (Etotal) of a closed system, which is equal to the sum of the total kinetic energy (KEtotal) and the gravitational potential energy (U) between the masses m1 and m2 separated by distance r.
Etotal=KEtotal+U, where U=−G(m1m2/r)
Which statement explains the reason why planets closer to the sun orbit faster than those that are farther away?

a) The value of KEtotal increases because U has a lesser negative value.
b) The value of G increases significantly the closer the planet is to the sun.
c) The value of Etotal increases significantly the closer the planet is to the sun.
d) The value of KEtotal increases because U has a larger negative value.

Answer 1

Answer:

Given

a

2

=13 and a

5

=25

We know that

a

n

=a+(n−1)d

∴a

2

=a+(2−1)d

⇒13=a+(2−1)d

⇒a+d=13 ...(i)

and a

5

=a+(5−1)d

⇒25=a+4d

⇒a+4d=25 ...(ii)

Now, subtracting (i) from (ii), we get

3d=12

⇒d=4

Now a+d=13 [from (i)]

⇒a=9

Hence, a

7

=a+6d=9+6(4)

⇒a

7

=33

Answer 2

The value of KE total increases because U has a larger negative value.

Therefore option D is the correct one.

• The energy of a system always remains conserved, it can only be transferred.
• So the sum of kinetic energy and potential energy remains constant, if one of them increases then the other one has to decrease and vice-versa.
• Since U = -Gm₁m₂/ r, where r is the distance between the two bodies. The planets that are closer to the Sun will have less r and therefore the overall value of Gm₁m₂/r will increase and U will be more negative as compared to the planets that have larger r (are far away from the sun).
• So if U is more negative then KE has to be more positive to keep the total energy constant.