Numford
Projectile motion

Complementary angles give the same range

θ and 90° - θ give the same range because sin 2θ = sin(180° - 2θ). At 30 m/s with g = 10 m/s², 15° and 75° both land 45 m away.

Projectile motion

Drag the launch arrow, or type the speed and angle. Slide the time to see the velocity at that moment. The dots are equal steps of time apart. Air resistance is ignored, as in NCERT.

Time of flight T5.8 s2u sin θ / g
Range R45 mu² sin 2θ / g
Max height H41.99 mu² sin²θ / 2g
Velocity at 1.45 s16.44 m/svx = 7.76, vy = 14.49, 61.81° up
Pathy = 3.73x - 0.0829x²y = x tan θ - gx² / (2u² cos²θ)

A ball thrown at 30 m/s and 75 degrees rises 41.99 m, flies for 5.8 s and lands 45 m away.

g (m/s²)
Show the formulas
Time of flight
T = 2u sin θ / g = 2 × 30 × sin 75° / 10 = 5.8 s
Range
R = u² sin 2θ / g = 30² × sin 150° / 10 = 45 m
Max height
H = u² sin²θ / 2g = 30² × sin²75° / (2 × 10) = 41.99 m
Longest range
At θ = 45°, R max = u² / g = 90 m. θ and 90° - θ give the same range.
Velocity
vx = u cos θ = 7.76 m/s, vy = u sin θ - gt = 14.49 m/s at t = 1.45 s, so v = 16.44 m/s
Path
y = x tan θ - gx² / (2u² cos²θ), here y = 3.73x - 0.0829x²
Working

Step by step

  1. Given u = 30 m/s at 15° and at 75°, with g = 10 m/s².
  2. At 15°: R = 30² × sin 30° / 10 = 45 m.
  3. At 75°: R = 30² × sin 150° / 10 = 45 m. sin 150° = sin 30°.
  4. Heights differ: H = 3.01 m at 15° and 41.99 m at 75°.
  5. Times differ too: T = 1.55 s and 5.8 s.
What the angle does

Range and height at 30 m/s

  • 15° R = 45 mH = 3.01 m, T = 1.55 s
  • 30° R = 77.94 mH = 11.25 m, T = 3 s
  • 45° R = 90 mH = 22.5 m, T = 4.24 s
  • 60° R = 77.94 mH = 33.75 m, T = 5.2 s
  • 75° R = 45 mH = 41.99 m, T = 5.8 s
Derivation

Why it works

  1. The range is R = u² sin 2θ / g.
  2. For the angle 90° - θ, sin 2(90° - θ) = sin(180° - 2θ) = sin 2θ.
  3. So θ and 90° - θ give exactly the same range, for example 15° and 75° or 30° and 60°.
  4. The steeper throw goes higher (H grows with sin²θ) and stays up longer (T grows with sin θ).
  5. The two angles are equally far from 45°, which is where the two paths meet as one.
Questions

Complementary angles, same range: quick answers

Why do complementary angles give the same range?

Because sin 2θ = sin(180° - 2θ) = sin 2(90° - θ). The two angles add to 90 degrees and give the same value of sin 2θ.

Which angle goes higher, 15° or 75°?

75°. With u = 30 m/s the heights are 3.01 m and 41.99 m.

Is there an angle with no partner?

Yes, 45 degrees is its own partner, and it gives the longest range of all.

Try your own throw