# Maths in a Box (15 Parts)

Synopsis
Series of video programmes on topics in mathematics. Each programme includes and event, a question and a solution.
1: A parachutist descends. A force decelerates the body (the parachute) while a force tries to accelerate it (gravity). How to make approximations and how the resultant of the forces can be equated to the product of mass and acceleration. How to use differential equations to answer questions such as the minimum height a parachutist can jump from without injury.
2: An aircraft gathers speed and lifts off the ground. The search for approximations becomes more complex. Graphical notation emphasises the interaction and variation of the forces. Shows a variety of aeroplanes, from single-engine trainers to Concorde, and finds that a single mathematical model could fit them all.
3: An arrow is accelerated by the bowstring. Shows how to make approximations to get to a model. Looks at how geometry can help solve relatively complex problems. Is landing on an aircraft carrier the archery problem in reverse? Uses geometric similarity to show show that the solution to one problem might help to solve another.
4: A child and a man pull a sledge across the sand. The child flies a kite. Vector algebra is used to show how the resultant of a number of forces acting on a body in equilibrium and non-equlibrium can be obtained and thereby equated to the product of mass and acceleration.
5: A balloon-powered toy car rushes across the floor losing mass as it goes. Newton’s second law is considered as the rate of change of momentum rather than F=ma. Examines how impulse relates to the change in momentum, and how the rate of this change equates to the resultant force. A small experiment illustrates jet propulsion.
6: A door, a lock gate, a heart valve opens and a gymnast flic-flacs down the gym. Once the principle of a moment has been defined, the lock gates and values can be represented by the same mathematics, to a first approximation.
7: At lift-off, the rocket burns fuel as it accelerates away, and the avalanche gathers snow as it pursues the skier, while flour slides down the baking tray. In each case mass changes. The problems range from finding the finite time for a rocket to lift off the ground to calculating whether the intrepid skier is engulfed.
8: The spinning top intrigues the professional engineer. Uses animated graphics to explain the principles and show how moments relate to the rate of change of angular momentum. Graphics show the use of vector products and their reasons for ‘precession’ of the top. Explains what difference size makes to an aircraft’s aileron flaps and how much force it takes to rotate toy spaceships.
9: Bikes lean over, car wheels react going round a bend, roads are cambered. Why? Centripetal acceleration was fundamental to Newton’s ideas about why the moon orbits the earth. Explores the idea that, on the basis of velocity being a vector of quantity, circular motion must imply a force or component of a force towards the centre of the circle.
10: The ice-hockey game and skiing head-on or sideways down the hill lead to questions about the physics and the mathematical models behind sliding and toppling. Investigates the relationship between sliding and the body’s velocity as an exercise in energy conservation.
11: From bridges to artificial hip joints, ours is a three-dimensional world. Shows the use and manipulation of vectors in modelling problems.
12: Real-life examples show that vectors may take a variety of guises other than a simple force. Looks at the more complex definitions of moments using vector geometry.
13: Before pendulums, there was only the sun to help measure time. Introduces the principles of simple harmonic motion, looking at how Galileo and his contemporaries began the analysis of pendulum motion. Then shows modern-day examples.
14: Investigates how the mathematical model can be increased in complexity through the introduction of damping. Here the homogeneous linear second order differential equation appears naturally by applying simple modelling techniques. Many examples from everyday life can be modelled by this type of equation.
15: Solution techniques for general linear second order differential equations are based on the idea of adding two independent solutions together. The physics of applying a time-varying force to a body, which fundamentally exhibits simple harmonic motion, shows the way in which mathematical models provide vital information and how the independent solutions are obtained. Resonance and its effects are investigated along with examples of naturally occurring and man-made phenomena.
Language
English
Country
Great Britain
Medium
Video; Videocassette. VHS. col. 21- 50 min. each
Year of production
1999
Availability
Sale; 1999 sale: £35.00 (+VAT +p&p) each part 1999 sale: £150.00 (+VAT +p&p) set of any 5 parts
Documentation
Accompanying booklet with each part.
Uses
A-level mathematics and 1st-year undergraduate engineering.
Subjects
Mathematics; Physics
Keywords
centripetal force; differential equations; equations; force; geometry; harmonic motion; mass; mathematical modelling; momentum; pendulums; vectors; velocity

### Sections

Title
Parachuting: moving bodies with constant mass
Synopsis
1: A parachutist descends. A force decelerates the body (the parachute) while a force tries to accelerate it (gravity). How to make approximations and how the resultant of the forces can be equated to the product of mass and acceleration. How to use diffe

Title
Take-off: moving bodies with constant mass
Synopsis
2: An aircraft gathers speed and lifts off the ground. The search for approximations becomes more complex. Graphical notation emphasises the interaction and variation of the forces. Shows a variety of aeroplanes, from single-engine trainers to Concorde, a

Title
Bows, arrow and aircraft carriers: moving bodies with constant mass
Synopsis
3: An arrow is accelerated by the bowstring. Shows how to make approximations to get to a model. Looks at how geometry can help solve relatively complex problems. Is landing on an aircraft carrier the archery problem in reverse? Uses geometric similarity

Title
Kites: modelling with vectors
Synopsis
4: A child and a man pull a sledge across the sand. The child flies a kite. Vector algebra is used to show how the resultant of a number of forces acting on a body in equilibrium and non-equlibrium can be obtained and thereby equated to the product of mas

Title
Impulse: moving bodies with variable mass
Synopsis
5: A balloon-powered toy car rushes across the floor losing mass as it goes. Newton's second law is considered as the rate of change of momentum rather than F=ma. Examines how impulse relates to the change in momentum, and how the rate of this change equa

Title
Doors, heart valves and flic flacs: moments
Synopsis
6: A door, a lock gate, a heart valve opens and a gymnast flic-flacs down the gym. Once the principle of a moment has been defined, the lock gates and values can be represented by the same mathematics, to a first approximation.

Title
Rockets and avalanches: moving bodies with variable mass
Synopsis
7: At lift-off, the rocket burns fuel as it accelerates away, and the avalanche gathers snow as it pursues the skier, while flour slides down the baking tray. In each case mass changes. The problems range from finding the finite time for a rocket to lift

Title
Spinning tops and aileroms: moments and angular momentum
Synopsis
8: The spinning top intrigues the professional engineer. Uses animated graphics to explain the principles and show how moments relate to the rate of change of angular momentum. Graphics show the use of vector products and their reasons for 'precession' of

Title
Bikes and cars: centripetal acceleration
Synopsis
9: Bikes lean over, car wheels react going round a bend, roads are cambered. Why? Centripetal acceleration was fundamental to Newton's ideas about why the moon orbits the earth. Explores the idea that, on the basis of velocity being a vector of quantity,

Title
Sliding and toppling: modelling forces
Synopsis
10: The ice-hockey game and skiing head-on or sideways down the hill lead to questions about the physics and the mathematical models behind sliding and toppling. Investigates the relationship between sliding and the body's velocity as an exercise in energ

Title
Modelling vectors
Synopsis
11: From bridges to artificial hip joints, ours is a three-dimensional world. Shows the use and manipulation of vectors in modelling problems.

Title
Vectors and moments
Synopsis
12: Real-life examples show that vectors may take a variety of guises other than a simple force. Looks at the more complex definitions of moments using vector geometry.

Title
Pendulum: simple harmonic motion
Synopsis
13: Before pendulums, there was only the sun to help measure time. Introduces the principles of simple harmonic motion, looking at how Galileo and his contemporaries began the analysis of pendulum motion. Then shows modern-day examples.

Title
Damping: simple harmonic motion
Synopsis
14: Investigates how the mathematical model can be increased in complexity through the introduction of damping. Here the homogeneous linear second order differential equation appears naturally by applying simple modelling techniques. Many examples from ev

Title
Resonance: simple harmonic motion
Synopsis
15: Solution techniques for general linear second order differential equations are based on the idea of adding two independent solutions together. The physics of applying a time-varying force to a body, which fundamentally exhibits simple harmonic motion,

### Production Company

Name

#### University of Leeds Media Services

Notes
see Leeds University Television

### Distributor

Name

#### Leeds University Television

Contact
Sally Popplewell (Sales supervisor)
Email
mediaservices@leeds.ac.uk
Web
http://mediant.leeds.ac.uk/vtcatalogue/ External site opens in new window
Phone
0113 343 2660
Fax
0113 343 2669