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Speaking the Same Language in Physics and Math
Marci K. Harvey
West Forsyth High SchoolClemmons, NC
Thanks To the Following:
• North Carolina Center for the Advancement of Teaching (NCCAT)
• Heather King, Math teacher• Daphne Marshall, Math teacher• Kurt Telford, Principal, West Forsyth
Today’s Objectives
• Demonstrate how students can make a connection between math and physics
• Compare solutions to typical problems with math and physics formulas
• Justify the benefit to students of teaching both methods
The Problem…
• Students do not make connections between physics concepts and math concepts, even if they take the courses concurrently!
• x = Vcost• y = Vsin()t-gt2/2
Speaking the Same Language…
Physics Unit: Freefall motion, kinematic equations
Math Unit: Parametric equations, quadratic equations
Problem: A ball is dropped from 100m on Earth, the moon, and Jupiter. How long does it take the ball to reach the surface at each location?
Forms of the Equations
• Math (parametric) Equations for Motion:y = ax2 + bx + c
• Physics Equations for Motion:yf = 1/2at2 + vit + yi
Parametric Solution…
gearth = -9.8 m/s2
gmoon = -1.6 m/s2
gjupiter = -26 m/s2
To set up calculator:• Turn “stat plot” OFF• Mode Par, Dot, Simul
• Y= X1T = 1 (moves object away from y-axis)
• Enter Y1T equation
– h(t) = -4.9t2 + Vot + ho
• Set “window”– Tmin = 0 s– Tmax = you decide– Tstep = 0.3 s recommended
100m
Parametric Solution…
gearth = -9.8 m/s2
gmoon = -1.6 m/s2
gjupiter = -26 m/s2
To set up calculator:
• Y= X2T = 2 for moon and X3T = 3 for Jupiter
• Enter Y2T and Y3T equations
• Graph and watch!• “Trace” to find time when
ball hits surface
100m
Physics Solution…
gearth = -9.8 m/s2
gmoon = -1.6 m/s2
gjupiter = -26 m/s2
Kinematic solution:
yf = yi + vit + ½gt2
0m = 100m + ½(-9.8m/s2)(t2)
t = 4.5 s for Earth
t = 11.2 s for moon
t = 2.8 s for Jupiter
100m
How about motion in two dimensions?
Physics Unit: 2D motion
Math Unit: Parametric Equations/Projectiles
Problem: An outfielder throws a softball 28 m/s at an angle of 55o from the ground. How long will the ball be in the air? Will it make it to the catcher, 80 m away?
Forms of the Equations
• Parametric Equations for Motion:x(t) = vtcos y(t) = vtsin – 1/2gt2 + h
• Physics Equations for Motion:
xf = xi + vt
yf = 1/2at2 + vit + yi
Parametric Solution…Parametric solution:
• Calculator settings– Degree mode
• Enter equations– X1T = 28T(cos55)
– Y1T = 28T(sin55) – ½(9.8)T2
• Set window:– Tmin = 0; Tmax = 5; Tstep = 0.1– Xmin = 0; Xmax = 100– Ymin = 0; Ymax = 50
55o
28 m
/s
Parametric Solution…Parametric solution:
• Graph and watch!• “Trace” to find time when ball
hits surface
Answer:
t = about 4.6 s
x = about 74 m
55o
28 m
/s
Physics Solution…Kinematic solution:
Vf = Vi + at
0 = 23 m/s + (-9.8m/s2)(t)
t = 2.3 s to the top of the parabola
t = 4.6 s for entire flight
x = vt = (16m/s)(4.6s)
= 73.6 m
It does not reach home.
55o
28 m
/s
Vx = (28 m/s)(cos55) = 16 m/s
Vy = (28 m/s)(sin55) = 23 m/s
Other Areas to Consider…
• Vector addition in physics
• Dot product and Law of Cosines in math
• x,v,a graphs/relationships in physics
• Integrals/derivatives in math
Questions? Final thoughts?
• Your ticket out the door!
• My email: [email protected]
• Phone: 336-712-4400