A rocket is launched from the surface of a planet. At the position shown in the diagram, The rocket is at a distance of 2 R from the planet (where R is the radius of the planet) and its speed is v = GM 2 R . At that point, the fuel runs out. Draw a graph to show how the speed of the rocket varies with the distance r from the centre of the planet as the rocket begins to fall backwards to the planet.

Given the distance of the rocket from the planet= 2 R The radius of the planet = R The velocity of the rocket at that point v = GM 2 R The distance travelled by a body under free fall is given by V 2 = U 2 + 2 gs where V , U are the final and initial velocities and s = distance covered. The distance can be represented as x = v 2 2 g - u 2 2 g This is a quadratic equation hence the shape of the curve is parabola. X-axis=velocity of the rocket Y- axis the distance from the planet. At the start of the fall the velocity is zero and will accelerate as it comes closer to the planet. The acceleration is also goes on increasing as it comes closer to the planet.

Prove that the total energy of earth (mass m ) as it orbits the Sum (mass M ) can be expressed as either E = - 1 2 mv 2 or E = - GMm 2 r where r is the radius of Earths' circular orbit. Calculate the energy numerically.

The kinetic energy of a moving body is given by E = 1 2 mv 2 ............................................................(1) where m = mass of the body and v = its speed. Now we can consider the earth as the satellite orbiting at a height equal to the radius of the orbit around the sun (= r ) The orbital velocity of the earth around the sun is v = GM r ....................(2) Substituting in the equation (1), we get E = - 1 2 m GM r 2 = - GMm 2 r Negative energy means It binds the two. Putting the standard values in the formula we have E = - G M m 2 ( r ) = - 6 . 67 × 10 - 11 N k g - 2 m - 2 × 2 × 10 30 k g × 5 . 97 × 10 24 k g 2 × ( 1 . 5 × 10 11 m + 6 . 4 × 10 6 + 6 . 96 × 10 8 ) where r = distance of the centres of the earth and the sun. E = - 5 . 3 × 10 33 J

The total energy of satellite during launch from the earth surface is E = - GMm 5 r , where R is the radius of the earth. Explain why this satellite will not escape the earth

Given the energy of the satellite = E = - GMm 5 r The satellite will not escape because the total energy is negative. This will bound the satellite with the earth. The kinetic energy of the satellite will be K = + GMm 5 r = 1 2 mv 2 or v = 2 GM 5 r This velocity is less than the escape velocity which is v e = GM r . Therefore the satellite will not escape from the earth.

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A rocket is launched from the surface of a planet. At the position shown in the diagram, The rocket is at a distance of $2 R$ from the planet (where $R$ is the radius of the planet) and its speed is $v = \sqrt{\frac{GM}{2 R}}$ . At that point, the fuel runs out.

Explain why the probe will eventually crash onto the surface of the planet.

Important Questions on Fields (HL)

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Diploma

IMPORTANT

Physics for the IB Diploma 6th Edition>Chapter 10 - Fields (HL)>Test yourself>Q 27

A rocket is launched from the surface of a planet. At the position shown in the diagram, The rocket is at a distance of $2 R$ from the planet (where $R$ is the radius of the planet) and its speed is $v = \sqrt{\frac{GM}{2 R}}$ . At that point, the fuel runs out.

Calculate in terms of $R$ , the maximum distance from the centre of the planet the rocket travels to.

Question Image

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Diploma

IMPORTANT

Physics for the IB Diploma 6th Edition>Chapter 10 - Fields (HL)>Test yourself>Q 27

Determine the speed with which the rocket crash onto the planet surface.
e rocket travels to. Question Image

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Diploma

IMPORTANT

Physics for the IB Diploma 6th Edition>Chapter 10 - Fields (HL)>Test yourself>Q 27

Draw a graph to show how the speed of the rocket varies with the distance $r$ from the centre of the planet as the rocket begins to fall backwards to the planet.

HARD

Diploma

IMPORTANT

Physics for the IB Diploma 6th Edition>Chapter 10 - Fields (HL)>Test yourself>Q 28

Prove that the total energy of earth (mass