Nuffield Free-Standing Mathematics Activity Galileo’s projectile model

1. Nuffield Free-Standing Mathematics Activity Galileo’s projectile model

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

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

4. x y Galileo’s projectile model Think about What can you say about bc, cd, and de? What does this tell you about the horizontal velocity of the ball and the horizontal distance covered by the ball? How could you check that the vertical distances are proportional to t2? a c e b d i o f g h l n Horizontal direction – the motion has constant speed. Vertical distance fallen is proportional to t2. Vertical direction – projectile accelerates at 9.8 ms–2.

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

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

7. Set up a model C B h R Experiment to validate Galileo’s model Think about What are the constants and variables? Constants • the horizontal velocity of the projectile after its launch from C • the acceleration is g downwards Variables • the time, tseconds, measured from the instant of launch • the height of the table h metres • the distance, Rmetres, the ball lands from the foot of the table

8. Analyse distance travelled • calculate the average velocity from time taken 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. Vary the release point A to vary the launch velocity Use talcum powder or salt on paper to find where the ball lands

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

10. Graph based on analysis Analyse gives Range Rmetres Velocity of projectionums-1 gives R= ut gradient Experiment to validate Galileo’s model Vertical motion downwards Horizontal motion

11. at time t (x, y) y x O uvert vvert uhoriz vhoriz Extension: projection at an angle to the horizontal In the horizontal direction, a = 0 In the vertical direction, a = –9.8 Find equations for vhoriz, x, vvertandyin terms ofuhoriz, uvert, t Sketch graphs of vhoriz, x, vvertandyagainstt

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

13. Galileo’s projectile model vvert y uvert 0 t 0 t In the vertical direction, a = –9.8 vvert = uvert– 9.8t y= uvertt– 4.9t2

14. 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.