Nuffield FreeStanding Mathematics Activity Galileos projectile model Nuffield

Nuffield Free-Standing Mathematics Activity Galileo’s projectile model © Nuffield Foundation 2011

Galileo’s projectile model How far will the ski jumper travel before he lands? How can you model the motion of the ski jumper?

Motion of a ball Height (y metres) Think about What assumptions are being made if the ball is modelled as a particle? Horizontal distance (x metres) Height (y metres) Horizontal distance (x metres) Time (t seconds) Think about Which feature of a distance-time graph represents speed? Time (t seconds)

Galileo’s projectile model Think about a What can you say about bc, cd, and de? What does this tell you about the x horizontal velocity of the ball and the horizontal distance covered by the ball? How could you check that the vertical distances are proportional to t 2? b o g l c i e d f h y Horizontal direction – the motion has constant speed. Vertical direction – projectile accelerates at 9. 8 ms– 2. n Vertical distance fallen is proportional to t 2.

The modelling cycle Real world Mathematics Set up a model Observe Define problem Analyse Validate Interpret Predict

Experiment to validate Galileo’s model You need: Think about What modelling assumptions will be made? A B C Assumptions • the ball is a particle height h metres • air resistance is negligible range R metres • the path of the projectile lies in a plane

Experiment to validate Galileo’s model Think about What are the constants and variables? Set up a model C B Constants h R • the horizontal velocity of the projectile after its launch from C • the acceleration is g downwards Variables • the time, t seconds, measured from the instant of launch • the height of the table h metres • the distance, R metres, the ball lands from the foot of the table

Experiment to validate Galileo’s model Use the equations for motion in a straight line with constant acceleration to predict : • how long it will take the ball to fall to the ground • the horizontal distance, R metres, it will have travelled Practical advice To estimate the velocity of the ball at launch: • assume the ball has constant velocity along BC. • time the ball travelling a measured distance along BC. • calculate the average velocity from distance travelled time taken Vary the release point A to vary the launch velocity Use talcum powder or salt on paper to find where the ball lands Analyse

Experiment to validate Galileo’s model Interpret Investigate how theoretical predictions compare with experimental results Graph based on analysis Range R metres Graph of experimental results Range R metres Velocity of projection u ms– 1 Why might there be discrepancies between the two graphs?

Experiment to validate Galileo’s model Vertical motion downwards Analyse gives Graph based on analysis Horizontal motion gives Range R metres gradient R = ut Velocity of projection u ms-1

Extension: projection at an angle to the horizontal vvert at time t (x, y) uvert vhoriz y O uhoriz x In the horizontal direction, a = 0 In the vertical direction, a = – 9. 8 Find equations for vhoriz, x, vvert and y in terms of uhoriz, uvert, t Sketch graphs of vhoriz, x, vvert and y against t

Galileo’s projectile model In the horizontal direction, a = 0 vhoriz = uhoriz x = uhoriz t x uhoriz 0 t

Galileo’s projectile model In the vertical direction, a = – 9. 8 vvert = uvert – 9. 8 t y = uvert t – 4. 9 t 2 vvert y uvert 0 t

Reflect on your work • What are the advantages of Galileo’s projectile model? • Do your experimental results validate the model? • Suggest some examples of motion which could not be modelled very well as projectiles.