Projectile motion: decomposition

Split the motion: constant velocity horizontally, constant acceleration g vertically. Two one-dimensional problems.

Projectile motion: decomposition0 min · Free lecture

In this lesson

  • Split the motion: constant velocity horizontally, constant acceleration g vertically. Two one-dimensional problems.

A projectile is two independent motions wearing one costume.

The projectile's velocity separates into uₓ = u cosθ (constant, no horizontal force) and u_y = u sinθ (changing at rate g). Position: x = uₓt, y = u_y t − ½gt².

The independence of x and y motions is the single idea that solves every projectile problem. Time is the shared clock — compute it from one axis, use it in the other.

x=ucos⁡θ t,y=usin⁡θ t−12gt2x = u\cos\theta\, t,\quad y = u\sin\theta\, t - \tfrac{1}{2}gt^2

Projectile coordinates

Worked example

For a fixed launch speed, the horizontal range of a projectile is maximum when the launch angle is:

  1. R = v0² sin 2θ / g
  2. R is maximum when sin 2θ = 1, i.e. 2θ = 90°, so θ = 45°

Answer: 45°

Transcript (0 min)
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Projectile motion: decomposition — FemtoLearn.
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Split the motion: constant velocity horizontally, constant acceleration g vertically. Two one-dimensional problems.

Frequently asked

Which component decides the time of flight?

The vertical one — the ground boundary lives there.

Printable notes (free)

Practice

Free · 4 questions with full solutions
  1. Q1 · Numerical · difficulty 2/5

    A projectile is launched with speed 15 m/s at an angle 70° above the horizontal. Taking g = 10 m/s², find its horizontal range.

    R=v02sin⁡2θgR = \frac{v_0^2 \sin 2\theta}{g}
  2. Q2 · MCQ · difficulty 1/5

    For a fixed launch speed, the horizontal range of a projectile is maximum when the launch angle is:

    • 30°
    • 45°
    • 60°
    • 90°
  3. Q3 · Numerical · difficulty 2/5

    A projectile is launched with speed 15 m/s at an angle 25°. Taking g = 10 m/s², find its maximum height above the launch point.

    H=v02sin⁡2θ2gH = \frac{v_0^2 \sin^2\theta}{2g}
  4. Q4 · Numerical · difficulty 2/5

    A projectile is launched with speed 25 m/s at an angle 15°. Taking g = 9.8 m/s², find its total time of flight.