Grade 10 physics project – Michel, ICS Mushrif. Projectile motion, air resistance and the Magnus effect, using free kicks.
keyboard shortcuts
Shortcuts: Space = kick | ←/→ = angle | ↑/↓ = speed | A/D = side spin | Z/V = aim | W/X = top spin
| R = air resistance on/off | S = save shot | C = clear saved | L = lesson mode | E = explain
| M = sound on/off | 1-4 = presets | K = 1D warm-up | ? = show/hide this
Kick settings
Famous free kicks (approximate!)
side view (y vs x) – wall 9.15 m out, 1.8 m tall. goal 2.44 m high. on a phone/tablet: drag from the ball to aim, longer drag = faster.
top view (curve) – goal is 7.32 m wide, wall is centred in front of it.
graphs: left = y vs x (the path), right = vy vs t. dashed = saved shots. remember: gradient of v–t = acceleration, area under v–t = displacement.
Warm-up: motion in 1D first. Before the free kick, one thing moving along a straight line, like in our kinematics notes. Set the starting position x0, initial velocity v0 and acceleration a, press Go and watch the three graphs being drawn. (+x = to the right.)
Examples from the kinematics worksheet:
max height H
range R
time of flight T
sideways at landing
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Data every 0.1 s:
t (s)
x (m)
y (m)
vx (m/s)
vy (m/s)
From 1D to 2D. In class we did kinematics in one dimension: v = v0 + at, Δx = v0t + ½at2, v2 = v02 + 2aΔx, free fall with a = −g, and reading displacement–time and velocity–time graphs.
The next step is a projectile: split the motion into a horizontal part (a = 0, constant velocity) and a vertical part (free fall, a = −9.8 m/s²) and use the same equations on each.
Six challenges. Do them in order and write your working in your book too. "Set up sliders" puts the numbers in for you. g = 9.8 m/s².
One kick, step by step, with the numbers from the sliders put into the equations (no air resistance, no spin – the “ideal” case first).
The ball (mass m = 0.43 kg, radius r = 0.11 m) has three forces on it: weight, drag and the Magnus force.
0. Recap from 1D kinematics (class notes): Δx = x − x0, v = v0 + at, Δx = v0t + ½at2, v2 = v02 + 2aΔx, free fall a = −g = −9.8 m/s².
1. No air (basic projectile = next step after 1D) – do the 1D equations twice: horizontally a = 0, vertically a = −g.
vx = v0 cosθ vy = v0 sinθ − g t
x = v0 cosθ · t y = v0 sinθ · t − ½ g t2
R = v02 sin 2θ / g H = v02 sin2θ / 2g T = 2 v0 sinθ / g
Check: 20 m/s at 45° gives R = 400/9.8 ≈ 40.8 m (turn drag off and spin to 0 to test it).
2. Drag (quadratic, because the ball is fast)
FD = ½ ρ CD A v2 (opposite to v)
ρ = 1.2 kg/m³, CD ≈ 0.25, A = πr2 = 0.038 m². Because it depends on v2 there is no simple formula any more.
3. Magnus force (the curl)
FM = ½ ρ CL A v2 in the direction of ω × v
CL ≈ rω/v (spin ratio), capped at 0.35. Side spin bends the ball left/right, top spin makes it dip, backspin makes it float. No spin + drag = the “knuckleball” (in real life it wobbles because the air flow goes turbulent unevenly – this sim doesn't model that bit).
How the sim works: it steps forward in time (Δt = 0.001 s): a = Fnet/m, update v, update position, repeat until y < 0 (midpoint method, a bit better than plain Euler).
Use g = 9.8 m/s². Show working. Give answers to 3 s.f. with units.
A ball is kicked at v0 = 20 m/s at θ = 30°. Find vx and vy at t = 0.
For the same kick (no air resistance) find the time of flight T, maximum height H and range R.
Find vy at t = 0.5 s. Is the ball going up or down?
Which angle gives the biggest range with no air? Prove it using R = v02 sin 2θ / g.
Use the sim: turn air resistance ON. Fill in the table.
θ (°)
R no drag (m)
R with drag (m)
% less
15
30
45
60
Does 45° still give the max range with drag? ____________
A free kick is 22 m from goal. The wall (1.8 m) is 9.15 m away. With v0 = 25 m/s and no air, what angle gets the ball over the wall but under the bar (2.44 m)? Check in the sim.
A kick is timed three times: 2.04 s, 1.92 s, 2.10 s (stopwatch reads to 0.01 s). Find the average time, its absolute uncertainty (half the range + reading error, 2 s.f.) and the percentage uncertainty.
Sketch the vy–t graph for a kick with no air. What does its gradient tell you? What does the area under it (up to the top) tell you?
Explain in your own words why side spin makes the ball curve (use the words Magnus force, pressure, velocity).
Answers (teacher): 1) 17.3, 10.0 m/s 2) 2.04 s, 5.10 m, 35.3 m 3) 5.10 m/s, up 4) 45° 7) 2.02 ± 0.10 s, about 5% 8) gradient = a = −9.8 m/s²; area = max height