Question

The maximum distance from the Earth to the Sun (at aphelion) is 1.521 1011 m, and the distance of closest approach (at perihelion) is 1.471 1011 m. The Earth's orbital speed at perihelion is 3.027 104 m/s. Ignore the effect of the Moon and other planets. (a) Determine the Earth's orbital speed at aphelion. m/s (b) Determine the kinetic and potential energies of the Earth–Sun system at perihelion. Kp = J Up = J (c) Determine the kinetic and potential energies at aphelion. Ka = J Ua = J (d)Is the total energy constant

Answer 1

:) *_* :3 ^-^ {.}{.}

Step-by-step explanation:

Answer 2

29274.93096 m/s

`2.73966* 10^(33)\ J`

`-5.39323* 10^(33)\ J`

`2.56249* 10^(33)\ J`

`-5.21594* 10^(33)`

Step-by-step explanation:

`r_p` = Distance at perihelion =
`1.471* 10^(11)\ m`

`r_a` = Distance at aphelion =
`1.521* 10^(11)\ m`

`v_p` = Velocity at perihelion =
`3.027* 10^(4)\ m/s`

`v_a` = Velocity at aphelion

m = Mass of the Earth = 5.98 × 10²⁴ kg

M = Mass of Sun =
`1.9889* 10^(30)\ kg`

Here, the angular momentum is conserved

`L_p=L_a\\\Rightarrow r_pv_p=r_av_a\\\Rightarrow v_a=(r_pv_p)/(r_a)\\\Rightarrow v_a=(1.471* 10^(11)* 3.027* 10^(4))/(1.521* 10^(11))\\\Rightarrow v_a=29274.93096\ m/s`

Earth's orbital speed at aphelion is 29274.93096 m/s

Kinetic energy is given by

`K=(1)/(2)mv_p^2\\\Rightarrow K=(1)/(2)* 5.98* 10^(24)(3.027* 10^(4))^2\\\Rightarrow K=2.73966* 10^(33)\ J`

Kinetic energy at perihelion is
`2.73966* 10^(33)\ J`

Potential energy is given by

`P=-(GMm)/(r_p)\\\Rightarrow P=-(6.67* 10^(-11)* 1.989* 10^(30)* 5.98* 10^(24))/(1.471*  10^(11))\\\Rightarrow P=-5.39323* 10^(33)`

Potential energy at perihelion is
`-5.39323* 10^(33)\ J`

`K=(1)/(2)mv_a^2\\\Rightarrow K=(1)/(2)* 5.98* 10^(24)(29274.93096)^2\\\Rightarrow K=2.56249* 10^(33)\ J`

Kinetic energy at aphelion is
`2.56249* 10^(33)\ J`

Potential energy is given by

`P=-(GMm)/(r_a)\\\Rightarrow P=-(6.67* 10^(-11)* 1.989* 10^(30)* 5.98* 10^(24))/(1.521* 10^(11))\\\Rightarrow P=-5.21594* 10^(33)`

Potential energy at aphelion is
`-5.21594* 10^(33)\ J`