Newton’s Laws

Newton’s first law In the absence of forces, a particle moves with constant Projectiles velocity. and Newton’s second law For any particle of mass m, the net force F on the particle is Charged Particles always equal to the mass m times the particle’s acceleration: F = ma Newton’s third law

If object 1 exerts a force F21 on object 2, then object 2

always exerts a reaction force F12 on object 1 given by

F 12 = −F 21

Newton’s Second Law Air resistance v Newton’s second law as a differential equation

d 2 r vˆ = v / | v | m 2 = F 2 dt f (v) = bv + cv = flin + fquad f = − f (v)vˆ or in the Cartesian coordinates w = mg d 2 x The physical origin of the terms: m 2 = Fx dt The linear term corresponds to the viscosity drag of the d 2 y m = F medium dt2 y The quadratic term describes the acceleration of the mass of d 2 z m = F air pushed by the projectile dt2 z for medium to fast speeds flin << fquad

Drag force at fast speed (Rayleigh’s equation) Motion in (x,y) plane Equations of motion with gravitational and drag forces r 1 F = − CρAvvr D 2 d 2 x m = F dt2 Dx C − drag coefficient (dimensionless constant) d 2 y ρ − air density m = −mg + FDy dt 2 A − projectile cross section (area) v − speed Imposing initial conditions the system of ordinary differential equations can be solved numerically using methods for solving ODE initial value problem Runge-Kutta method normally works well for the system

1 r 1 r r FD = − CρAvv = −Dvv Drag coefficient C 2 Terminal speed (approx.) 1 mg = CρAv2 drag coefficient C depends on an object shape and can be 2 determined by wind tunnel measurements. speed speed distance For many objects it can be approximated by a value object within 0.05 – 1.0 (m/s) (mph) (m) 95% shot 145 316 2500 sky diver 60 130 430 baseball 42 92 210 basketball 20 44 47 raindrop 7 15 6 parachutist 5 11 3

r 1 Aerodynamic drag crisis F = − C(v)ρAvvr D 2 References

Generally C depends on speed v (aerodynamic drag crisis) R. Adair The Physics of baseball. New York, Harper and Example for baseball: Frohlich Am J. Phys. 52, 325 (1984). Row (1990) C. Lindsey et al The Physics and Technology of Tennis, USRSA (2004) T. P. Jorgensen The Physics of Golf, American Institute of Physics (1999) J. Wesson The Science of Soccer, Institute of Physics Publishing (2002)

vd R = v – velocity, d – diameter, w w – kinematic viscosity of air

vˆx = vx / | v | Drag force in (x,y) plane System of equations for calculations vˆy = vy / | v | d 2 x = −Dvv / m r 1 r r 2 x FD = − CρAvv = −Dvv dt 2 2 d y 1 = −g − Dvvy / m where D = CρA dt2 2

since air density varies with altitude, one may approximate it as 2 2 2 FDx = −Dv cos(θ ) = −Dvvx = −Dvx (vx + vy ) ρ = ρ0 exp(− y / y0 ) 2 2 2 where ρ0 is the density as sea level (y=0). FDy = −Dv sin(θ ) = −Dvvy = −Dvy (vx + vy ) the coefficient y0 is about 10,000 m

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30 A good test case Initial conditions no air resistance air resistance (same air density) speed 800 m/s 25 air resistance with ρ=ρ exp(-y/y ) Attention! angle 450 0 0 example drag coeff. C=0.2 A good way to test your computer code. 3 20 air density 1.25kg/m R =0.1 m The system has an analytic solution for D=0 projectile M =100 kg projectile x = x + v cos(ϑ )t v = v cos(ϑ ) 15

0 0 0 x 0 0 2 y = y + v sin(ϑ )t − gt / 2 vy = v0 sin(ϑ0 ) − gt

0 0 0 (km) altitute 10 and the equation of the path is 5 sinθ g x2 y = y + x 0 − 0 cosθ 2 v2 cos2 θ 0 0 0 0 0 10203040506070 distance (km)

10 r r r Effect of spin (Magnus force) FM = S(v)ω × v Golf ball no air resistance air resistance speed 70 m/s 8 angle 9 degrees topspin example

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4 altitute (m) altitute backspin

2 F common approximation M = S ω v m 0 0 0 50 100 150 200 coefficients for S0 can be found is elsewhere distance (m)

Effect of wind (parallel to the ground) Golf ball no air resistance air resistance 30 speed 70 m/s let w is the speed of wind, then with backspin angle 9 degrees example + w corresponds to tailwind - w corresponds to headwind

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v = u w d x D x x m = − (u w)v dt 2 m x m altitute (m) altitute vy = uy d 2 y D 10 2 2 v = (u w) + u 2 = −g − uyv x m y dt m

0 0 50 100 150 200 250 distance (m)

3 Lorentz force on a charged particle Possible projects

F = q(E + v × B) Projectile motion in sport Motion in electric and magnetic fields for motion in crossed fields Laser cooling and trapping

E ⊥ B E = (Ex , Ey , 0) B = (0, 0, Bz )

d 2 x 2 d x Ex m 2 = Ex + qBzvy = + ωv dt dt2 m y 2 2 d y qB d y E y m 2 = Ey − qBzvx ω = = − ωv dt m dt2 m x 2 d z d 2 z m 2 = 0 = 0 dt dt2

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