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

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

Horizontal distance ( x metres) Height ( y metres) Time ( t seconds) Height ( y metres) Time ( t seconds) Horizontal distance ( x metres) 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? Motion of a ball

Galileo’s projectile model x y Horizontal direction – the motion has constant speed. Vertical direction – projectile accelerates at 9.8 ms –2. Vertical distance fallen is proportional to t 2. a o b c d e f h i g l n 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 t 2 ?

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

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

Constants the horizontal velocity of the projectile after its launch from C the acceleration is g downwards R h B C 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 Set up a model Think about What are the constants and variables? Experiment to validate Galileo’s model

Analyse 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 Experiment to validate Galileo’s model

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

Graph based on analysis Range R metres Velocity of projection u ms -1 Vertical motion downwards gives Horizontal motion R = ut gives gradient Analyse

Extension: projection at an angle to the horizontal x y O u horiz u vert v horiz v vert ( x, y ) at time t Find equations for v horiz, x, v vert and y in terms of u horiz, u vert, t In the horizontal direction, a = 0 In the vertical direction, a = –9.8 Sketch graphs of v horiz, x, v vert and y against t

x = u horiz t x t 0 v horiz = u horiz u horiz v horiz t 0 In the horizontal direction, a = 0 Galileo’s projectile model

v vert = u vert – 9.8 t 0 t v vert u vert y = u vert t – 4.9 t 2 t y 0 In the vertical direction, a = –9.8 Galileo’s projectile model

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.