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WINSTONCH [101]
3 years ago
11

Explain the relationship between mass and change in velocity. You may write

Physics
1 answer:
Klio2033 [76]3 years ago
8 0
It makes me cracked at Fortnite
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A) Orbital speed: v=\sqrt{\frac{GM}{R}}

B) Kinetic energy: K= \frac{GmM}{2R}

D) The orbital period is T=\frac{2\pi}{\sqrt{GM}}R^{3/2}

F) The angular momentum is L=m\sqrt{GMR}

G) Exponent of radial dependence:

Speed: -1/2

Kinetic energy: -1

Orbital period: 3/2

Angular momentum: 1/2

Explanation:

A)

We know that for a satellite in circular orbit around a planet of mass M, the gravitational force between the satellite and the planet is

F=G\frac{mM}{R^2}

where m is the mass of the satellite.

This force provides the centripetal force needed for the circular motion, which is

F=m\frac{v^2}{R}

where v is the orbital speed.

Since the gravitational force provides the centripetal force, we can equate the two expressions:

G\frac{mM}{R^2}=m\frac{v^2}{r}

And solving for v, we find

v=\sqrt{\frac{GM}{R}}

B)

The kinetic energy of an object is given by

K=\frac{1}{2}mv^2

where

m is the mass of the object

v is its speed

In this problem,

m is the mass of the satellite

v=\sqrt{\frac{GM}{R}} is the speed of the satellite (found in part A)

Substituting, we find an expression for the kinetic energy of the satellite:

K=\frac{1}{2}m(\sqrt{\frac{GM}{R}})^2 = \frac{GmM}{2R}

D)

The orbital speed of the satellite can be rewritten as the ratio between the distance covered during one orbit (the circumference of the orbit) divided by the period of revolution:

v=\frac{2\pi R}{T}

where

2\pi R is the circumference of the orbit

T is the orbital period

We already found that the orbital speed is

v=\sqrt{\frac{GM}{R}}

Substituting into the equation,

\sqrt{\frac{GM}{R}}=\frac{2\pi R}{T}

And making T the subject,

T=\frac{2\pi R}{\sqrt{\frac{GM}{R}}}=\frac{2\pi}{\sqrt{GM}}R^{3/2}

F)

The angular momentum of an object is defined as

L=mvr

where

m is the mass of the object

v is its speed

r is the radius of the orbit

For the satellite here we have

m (mass of the satellite)

v=\sqrt{\frac{GM}{R}} (orbital speed)

R (orbital radius)

Substituting,

L=m\sqrt{\frac{GM}{R}}R=m\sqrt{GMR}

G)

First, we rewrite the list of expressions for the different quantities that we found:

Orbital speed: v=\sqrt{\frac{GM}{R}}

Kinetic energy: K= \frac{GmM}{2R}

Orbital period: T=\frac{2\pi}{\sqrt{GM}}R^{3/2}

Angular momentum: L=m\sqrt{GMR}

Now we observed the dependence of each quantity from R:

Orbital speed: v\propto R^{-1/2}

Kinetic energy: K \propto R^{-1}

Orbital period: T \propto R^{3/2}

Angular momentum: L \propto R^{1/2}

So the exponent of the radial dependence of each quantity is:

Speed: -1/2

Kinetic energy: -1

Orbital period: 3/2

Angular momentum: 1/2

Learn more about circular motion:

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7 0
3 years ago
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