Module 5: Advanced Mechanics

Split the motion into horizontal and vertical parts before any number: the horizontal velocity stays the same, the vertical one changes at g.

A lesson for this module

Quick checks: multiple choice

Question 1

A netball is thrown at an angle above the horizontal. Ignoring air resistance, how does the horizontal component of its velocity change during the flight?

Question 2

A cricket ball is hit high into the air. Ignoring air resistance, what is its acceleration at the highest point of its path?

Question 3

A ball rolls off the edge of a flat roof 20.0 m above level ground with a horizontal velocity of 6.0 m s⁻¹. Ignoring air resistance, how far from the base of the building does it land?

Question 4

A soccer ball is kicked from level ground at 20.0 m s⁻¹, 30.0° above the horizontal. Ignoring air resistance, what is its time of flight?

Question 5

A basketball is thrown at 15.0 m s⁻¹, 60.0° above the horizontal. Ignoring air resistance, how far above its release point does it rise?

Question 6

A ball is thrown from the edge of a roof 12.0 m above level ground at 18.0 m s⁻¹, 35.0° above the horizontal. Ignoring air resistance, how far from the base of the building does the ball land?

Question 7

On a level school oval, one student drops a tennis ball from a height of 1.5 m at the same instant as another student throws an identical ball horizontally from the same height at 8 m s⁻¹. Ignoring air resistance, which statement is correct?

Question 8

Two identical shot puts are launched from ground level at the same speed, one at 25° and one at 65° above the horizontal. Ignoring air resistance, how do their ranges compare?

Question 9

A stone is thrown from the top of a sea cliff 25.0 m above the water at 12.0 m s⁻¹, 40.0° above the horizontal. Ignoring air resistance, with what speed does it hit the water?

Question 10

The graph shows the vertical velocity of a ball thrown from level ground, taking upwards as positive. Ignoring air resistance, what is the greatest height the ball reaches?

Question 11

Range of the ball for each launch angle
Launch angle (°)	Range (m)
25	2.71
30	2.79
35	2.83
40	2.81
45	2.74
50	2.62
55	2.44

A student fires a ball from a launcher clamped to a bench, 0.90 m above the floor, at the same speed each time, and measures where it lands on the floor. Her results are in the table. Why is the greatest range at an angle below 45°?

Question 12

To test the relationship between launch angle and range, a student measures the range of a marble fired once at each angle from a launcher. Which change most improves the reliability of her results?

Question 13

A ball is launched from level ground at 25.0 m s⁻¹, 50.0° above the horizontal. Ignoring air resistance, what is its speed 1.50 s after launch?

Question 14

A stone is thrown horizontally from the top of a cliff 20 m above a beach and lands 30 m from the base of the cliff. Ignoring air resistance, what was its launch speed?

Question 15

A ball is kicked from level ground at 20.0 m s⁻¹, 53.0° above the horizontal, towards a wall 25.0 m away. Ignoring air resistance, how high above the ground is the ball when it reaches the wall?

Question 16

A car drives around a flat circular roundabout at constant speed. What is the direction of its acceleration?

Question 17

What provides the centripetal force on a car turning a corner on a flat road?

Question 18

A 1200 kg car drives around a flat bend of radius 50 m at a constant 15 m s⁻¹. What is the size of the sideways friction force on its tyres?

Question 19

The drum of a washing machine has a radius of 0.25 m and spins at 1200 revolutions per minute. What is the speed of a sock pressed against the inside of the drum?

Question 20

A wind turbine near Bungendore turns through 3 complete revolutions in 12 s at a steady rate. What is its angular velocity?

Question 21

The same turbine turns through 3 complete revolutions in 12 s, and each blade is 40 m long from the axle to its tip. What is the speed of a blade tip?

Question 22

A curve on a test track has a radius of 80.0 m and is banked at 15.0° to the horizontal. At what speed can a car take the curve with no sideways friction from the road?

Question 23

A car rounds a banked curve at exactly its design speed, so no sideways friction acts. What provides the centripetal force?

Question 24

A puck tied to a string moves in a horizontal circle on a frictionless air table. Its speed is doubled and the radius of the circle is halved. By what factor does the tension in the string change?

Question 25

A student pulls on the end of a spanner 0.30 m long with a force of 50 N, at an angle of 60° to the spanner's handle. What torque does the force produce about the nut?

Question 26

Why is a door handle fitted on the edge of the door furthest from the hinges?

Question 27

A stone on a string is whirled in a horizontal circle at constant speed. How much work does the tension in the string do on the stone in one revolution?

Question 28

A student whirls a 0.20 kg mass on a string in a horizontal circle of fixed radius and measures the tension for several speeds. Her graph of tension against speed squared is shown. What radius did she use?

Question 29

The tyres of a 1500 kg car can exert a sideways friction force of up to 6000 N on a flat road. What is the greatest speed at which the car can take a flat bend of radius 40 m?

Question 30

In a fairground rotor, riders stand against the inside wall of a cylinder of radius 2.5 m. The cylinder spins and the floor drops away. The wall's friction on a rider can be at most 0.40 times the normal force the wall exerts. What is the least speed of the wall for which riders do not slide down?

Question 31

A car takes a banked curve faster than the curve's design speed but does not skid. In which direction does the friction force from the road act on the tyres?

Question 32

What is meant by the gravitational field strength at a point?

Question 33

Using the data sheet values for Earth, what is the gravitational field strength at a height of 6.371 × 10⁶ m above Earth's surface?

Question 34

A 500 kg satellite orbits 4.0 × 10⁵ m above Earth's surface. Using the data sheet values for Earth, what is the gravitational force on it?

Question 35

An Earth-observation satellite is in a circular orbit 7.00 × 10⁵ m above Earth's surface. Using the data sheet values for Earth, what is its orbital speed?

Question 36

A geostationary satellite has an orbital period of 24.0 h. Using the data sheet values for Earth, what is its height above Earth's surface?

Question 37

Planet X orbits a star at 4 times the orbital radius of planet Y, which orbits the same star. How does the orbital period of X compare with that of Y?

Question 38

A 1200 kg satellite is moved from a circular orbit of radius 7.0 × 10⁶ m to a circular orbit of radius 4.2 × 10⁷ m. Using the data sheet values for Earth, how much does its total energy increase?

Question 39

The Moon has a mass of 7.3 × 10²² kg and a radius of 1.7 × 10⁶ m. What is the escape velocity from its surface?

Question 40

A comet moves around the Sun in a long elliptical orbit. Where in its orbit is its speed greatest?

Question 41

Astronauts on a space station orbiting about 400 km above Earth float freely in the cabin. Why?

Question 42

Orbital radius and period of three moons
Moon	Orbital radius (m)	Period (days)
A	2.25 × 10⁸	1.50
B	3.57 × 10⁸	3.00
C	5.66 × 10⁸	6.00

An astronomer measures the orbits of three moons of a distant planet. The orbits are close to circular, and the table gives them. What is the mass of the planet?

Question 43

Why is the gravitational potential energy of a satellite, U = −GMm ÷ r, a negative quantity?

Question 44

A satellite in a low circular orbit meets a thin trace of atmosphere, and drag slowly lowers it into smaller circular orbits. What happens to its orbital speed and its total energy?

Question 45

A space station orbits 420 km above Earth's surface. Using the data sheet values for Earth, what is its orbital period?

Question 46

A 400 kg probe is in a circular orbit of radius 1.0 × 10⁷ m around Earth. Using the data sheet values for Earth, what is the least energy that must be added for it to escape from Earth's gravity?

Question 47

A ball is launched from level ground at 20.0 m s⁻¹ at an angle above the horizontal. At the highest point of its path its speed is 8.00 m s⁻¹. Ignoring air resistance, how high above the ground does the ball rise?

Question 48

A coin placed 8.0 cm from the centre of a level turntable just begins to slide when the turntable turns at 45 revolutions per minute. An identical coin is placed 18 cm from the centre on the same surface. At what rate of turning does it just begin to slide?

Question 49

Earth (mass 6.0 × 10²⁴ kg) and the Moon (mass 7.3 × 10²² kg) are 3.84 × 10⁸ m apart, centre to centre. A spacecraft travels along the line joining their centres. How far from Earth's centre is the point where the net gravitational field of Earth and the Moon is zero?

Section II written questions

A cricket ball hit from above the groundSection II style · 8 marks

A batter hits a cricket ball when it is 1.10 m above level ground. The ball leaves the bat at 28.0 m s⁻¹, 32.0° above the horizontal. Ignore air resistance in parts (a) and (b).

(a) Calculate the greatest height above the ground that the ball reaches. [2]

(b) Calculate the horizontal distance the ball travels before it hits the ground. [3]

(c) In a real hit, air resistance acts on the ball. Explain how air resistance changes the ball's flight compared with your answers to (a) and (b). [3]

0 words · about 50 to 80 words

Moving a satellite to geostationary orbitSection II style · 7 marks

A 2000 kg communications satellite is launched into a circular parking orbit 3.00 × 10⁵ m above Earth's surface. It is later moved to a circular geostationary orbit, which has a period of 24.0 h. Earth's mass and radius are on the data sheet.

(a) Calculate the radius of the geostationary orbit. [2]

(b) Calculate the energy that must be supplied to move the satellite from the parking orbit to the geostationary orbit. [3]

(c) Explain why a geostationary orbit suits a satellite that relays signals between fixed dish antennas on the ground. [2]

0 words · a sentence or two

A banked exit rampSection II style · 5 marks

A highway exit ramp is a curve of radius 120 m, banked at 18.0° to the horizontal. Cars of mass about 1500 kg use the ramp.

(a) Calculate the speed at which a car can take the ramp with no sideways friction from the road. [2]

m s⁻¹

(b) A car takes the ramp at 30 m s⁻¹ without skidding. Explain how the forces on the car allow it to follow the curve at this speed. [3]

0 words · about 50 to 80 words

Whirling a stopper through a glass tubeSection II style · 8 marks

A student ties a 0.025 kg rubber stopper to a string that passes through a vertical glass tube. A 0.200 kg mass hangs from the lower end of the string, so the tension stays the same. She whirls the stopper in a horizontal circle above her head, keeping a paper-clip marker just below the tube so that the radius stays fixed, and times 10 revolutions with a hand-held stopwatch. She does three trials at each radius. Her mean results are in the table.

Mean time for 10 revolutions at each radius
Radius (m)	Time for 10 revolutions (s)
0.40	4.49
0.60	5.50
0.80	6.35
1.00	7.10

(a) Identify the independent variable, the dependent variable and one controlled variable in this investigation. [2]

0 words · a sentence or two

(b) Calculate the speed of the stopper when the radius is 0.80 m. [2]

m s⁻¹

0 words · about 80 to 120 words

Weighing a planet by its moonSection II style · 5 marks

Astronomers observe a moon moving in a nearly circular orbit of radius 3.50 × 10⁸ m around a distant planet. The moon's orbital period is 4.20 days.

(a) Explain how Newton's law of universal gravitation and the model of uniform circular motion lead to r³ ÷ T² = GM ÷ 4π² for a moon in a circular orbit. [3]

0 words · about 50 to 80 words

(b) Calculate the mass of the planet. [2]

Loosening a wheel nutSection II style · 4 marks

A mechanic tries to loosen a tight wheel nut with a wrench. She pushes on the end of the handle, 0.45 m from the centre of the nut, with a force of 220 N at 70° to the handle. The nut does not turn.

(a) Calculate the torque her push applies to the nut. [2]

N m

(b) Explain one way the mechanic could apply a larger torque to the nut without pushing harder. [2]

0 words · a sentence or two

A satellite's orbit decaysSection II style · 5 marks

A 950 kg Earth-observation satellite is in a circular orbit 5.00 × 10⁵ m above Earth's surface. Over several months, drag from the very thin upper atmosphere slowly lowers it into smaller circular orbits. Earth's mass and radius are on the data sheet.

(a) Calculate the satellite's orbital speed at 5.00 × 10⁵ m. [2]

m s⁻¹

(b) Drag acts against the satellite's motion, yet its orbital speed increases as its orbit decays. Explain why. [3]

0 words · about 50 to 80 words