11 OUT INTO SPACE Rhythms of the heavens Review knowledge and understanding of planetary motions Learn Kepler’s laws of planetary motion Analyse planetary orbit data to determine Kepler’s 3 rd law empirically

Q1. What is the significance of each of the following: 23 hours 56 min ; 24 hours ; 24 hours 49 min ; 28 days ; days Q2. Why do celestial bodies move across the sky from East to West each night? Are there any that don’t? Why not? Q3. Why are the planets sometimes referred to as “wandering stars”? Q4. Why does Mars show retrograde motion when tracked over several months? Q5. Why do we see different constellations at different times of year? Q6. Why is Halley’s comet only seen once every 76 years? Q7. Why do we only see one side of the Moon from Earth?

Q1. Use Kepler’s 3 rd law (T 2 = constant x R 3 ) to predict the time it takes Saturn to go once round the Sun: Mean distanceOrbital from Sun (AU)period (years) Venus Earth Mars Jupiter Saturn9.54? Q2. Can you justify Kepler’s 2 nd Law (p36) in terms of conservation of energy? (Hint: For a planet orbiting the Sun, the sum of KE + GPE will be constant)

Uniform circular motion Explain circular motion in terms of centripetal force, identifying its origin Derive and use the equations of circular motion

Q1. Calculate the angular velocity (in Radians s -1 ) for the following examples of uniform circular motion: (a)The seconds hand on a watch (b)The minute hand on a watch (c)The hour hand on a watch Q2. Calculate the centripetal acceleration of a point on the equator of the Earth due to the rotation of the Earth. Data: radius of Earth = 6.36 x 10 6 m Q3. The measured acceleration due to gravity at the Equator is about 9.78 ms -2, while at the poles it is 9.83 ms -2. Explain these observations in light of your answer to Q2. Q4. You swing a bucket of water, mass 5 kg, on a 0.9 m long string in a vertical circle, so that the bucket completes two complete revolutions in 3 seconds. (a)Calculate the centripetal acceleration and centripetal force acting on the bucket. (b)What is the tension in the string when the bucket is at (i) the top of the circle and (ii) the bottom of the circle?

A feeling of “weightlessness” will arise if the centripetal force is equal to the weight of the object. If a “vomit comet” follows a circular arc of radius 1000 m to achieve “weightlessness” of the occupants, what is the required speed?

Newton’s Law of Universal Gravitation Understand Newton’s reasoning in deriving NLUG Derive Kepler’s 3 rd law from a consideration of NLUG and centripetal force Investigate stability of orbital systems

The Moon takes about 28 days to orbit the Earth. The Moon is about km from the centre of the Earth. Q1. Calculate the angular velocity (in Radians s -1 ) of the Moon as it orbits the Earth. Q2. Calculate the centripetal acceleration of the Moon (ms -2 ). Q3. What provides the centripetal force that keeps the Moon orbiting the Earth? Q4. What is the acceleration of an object at the surface of the Earth (ms -2 )? Q5. What provides the force that causes this acceleration? Q6. Newton’s hypothesis was that it was the Earth’s gravity, “diluted” with distance, that provided the force to keep the Moon in orbit. Complete the table below and test to see how g decreases with distance. (Hint: Test g = k/d n, where k is a constant and n is an integer, for different values of n. Radius of Earth = 6360 km) LocationDistance from Earth’s centre (km) g (ms -2 ) Surface of Earth Moon

NLUG proves Kepler #3 ! 1. Write down an expression for the gravitational force between the Sun and a planet (Eqn.1) 2. Write down an expression for the angular velocity of a planet in terms of its period T (time to go round Sun) (Eqn.2) 3. Write down an expression for the centripetal force on the planet in terms of ω. (Eqn.3) 4. Substitute Eqn.2 into Eqn.3 to get Eqn Equate Eqn.4 to Eqn Manipulate algebraically to show that T 2 = constant x R 3 My 3 rd law has me baffled! Then try a dose of NLUG and see your problems vanish!

Problem solving with NLUG (Take G = 6.7 x N m 2 kg -2 ) Q1. (a) Calculate the force of gravitational attraction that you exert on the person next to you. (1 stone = 6.4 kg) Q1. (b) How big is the force of gravitational attraction that s/he exerts on you? Q2. (a) What force does the Earth exert on a 1 kg mass on its surface? Q2. (b) What force does Mars exert on this 1 kg mass? Data:Earth mass = 6 x kg; Mars mass = 6.4 x kg; Closest Earth-Mars approach distance = 7.8 x m Q3. In light of your answers to Q2, should you be worried by a planetary alignment?

Gravitational fields Explain that a gravitational field is a region in space where a mass feels a force due to another mass Describe and explain the variation of the radial field with distance from a massive body Use field considerations in calculations of satellite orbit radii, asteroid masses etc.

Geostationary orbits (Take G = 6.7 x N m 2 kg -2 ) A satellite placed in a geostationary orbit remains over the same point on the equator as the Earth rotates. Q1. Use the equation you derived for Kepler’s 3 rd law to calculate the correct orbit radius and height above the Earth’s surface for such a satellite. Q2. Calculate the strength of the Earth’s gravitational field at this location. Q3. Using a diagram or otherwise, calculate the minimum number of geostationary satellites needed for complete coverage of the Earth. Data: Mass of Earth = 6 x kg Radius of Earth = 6360 km

Determining asteroid masses (Take G = 6.7 x N m 2 kg -2 ) A probe is in orbit 300 km above the centre of a spherical asteroid. It takes 314 minutes to make one orbit around the asteroid. Q1. Show that the orbital speed of the probe is about 100 ms -1. Q2. Hence show that the centripetal acceleration of the probe is about ms -2. Q3. Explain why the gravitational field strength due to the asteroid at the probe’s orbit radius must be Nkg -1. Q4. Use your answer to Q3 to show that the asteroid’s mass is about 4.4 x kg.

What will happen when the moving spacecraft docks with the stationary one? Why does the gun recoil?..and which member of the crew needs to study Chapter 11?

Momentum Recap principle of momentum conservation Apply to a solve problems for a range of situations where bodies interact

BEFORE 10kg 2 ms -1 5 kg not moving AFTER 1 ms kg 5 kg 6 ms -1 Q1. Is momentum conserved in this collision? Q2. Can you nevertheless explain why this collision violates a fundamental law of physics, and therefore would never take place?

Q1. Predict what will happen when a light object travelling to the right strikes a stationary object that is 10 times more massive. (You can explore this with physics-online/virtual air track) Q2. Suppose the mass of the heavy ball is increased until it is 100 times more massive than the light ball. What happens now? (You can explore this with Modellus/Activity 160S) Q3. Explain the relevance of the above to “gravitational sling shotting” of spacecraft by a planet.

Impulse Impulse is a measure of the change of momentum of a body when a resultant force acts for a certain time. Starting with Newton’s 2 nd law (F = ma) and a relevant suvat equation, derive an expression for the change in momentum of a body in terms of the force acting on it and the time for which the force acts.

Impulse Show that impulse = change in momentum = F x Δt Review Newton’s 3 rd law and its applications Solve problems involving impulse considerations, including ball sports and rocket motion examples

How does FΔt = mΔv help us understand the following...

(N) (s) Force-time graph for tennis racket hitting a ball How can we calculate change in momentum from the graph?

How does FΔt = mΔv help us understand the following... Note that when two objects collide/interact, the change in momentum of one body is equal in size but opposite in direction to the change in momentum of the other

Impulse questions Q1. A ball of mass 0.06 kg moving at 15 ms -1 hits a wall at right angles and bounces off along the same line at 10 ms -1. (a)What is the magnitude of the impulse of the wall on the ball? (b)What is the magnitude of the impulse of the ball on the wall? (c)If the ball is in contact with the wall for 3 x seconds, estimate the average force on the ball. (d)Sketch a graph showing how the force on the ball would vary with time during the impact. (e)Is it possible to calculate the velocity with which the wall “moves away” after the impact? What information would you need? Q2. The ion engine on a spaceship in deep space produces an accelerating force of 20 N for 100 seconds. (a)Calculate the impulse provided by the ion engine. (b)If the spaceship has a mass of kg and is initially travelling at 2000 ms -1, what is its final velocity after the 100 second thrust phase? (c)What do you conclude about the usefulness of ion drives in spacecraft propulsion?

Gravitational potential Explain the meaning of the term, and how it is calculated for radial fields Describe the relationship between field strength and potential gradient Use kinetic-potential energy exchange considerations to explain escape velocity, cometary orbits, energetics of satellite orbits etc.