A.1 · IB Physics
Projectiles with and without air resistance
Launch two identical projectiles at once: one in a vacuum and one feeling a drag force proportional to v². Compare their paths, range, maximum height, time of flight and velocity components as they fly.
| Vacuum | With drag | |
|---|---|---|
| Range R | ||
| Max height H | ||
| Time of flight T | ||
| vx now | ||
| vy now | ||
| |v| now |
t = · Terminal speed vt = √(mg/k): · Drag range ÷ vacuum range:
Vacuum: ax = 0, ay = −g
Drag: F = k v², opposite to v
ax = −(k/m)|v| vx
ay = −g − (k/m)|v| vy
Vacuum range R = u² sin 2θ / g
g = 9.8 m s⁻²
Drag: F = k v², opposite to v
ax = −(k/m)|v| vx
ay = −g − (k/m)|v| vy
Vacuum range R = u² sin 2θ / g
g = 9.8 m s⁻²
Try this
- Set k to 0: the two paths coincide. Check that the range matches u² sin 2θ / g.
- With u = 40 m s⁻¹ and k = 0.010 × 10⁻³… try k = 10.0 × 10⁻³ kg m⁻¹, then vary θ to find the angle giving the largest drag range. Is it above or below 45°?
- Keep k fixed and double the mass. Watch the terminal speed and the drag range: why does a heavier ball suffer less from the same drag force?
Exam tip: when sketching a path with air resistance, draw it lower and shorter than the parabola and make the descent steeper than the ascent, because the horizontal velocity component falls throughout the flight.