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2.6

Projectile motion

Your guide: Sir HamzaBelieves every formula has a story, and units never lie.

Projectile motion

A batter hits a ball. It rises, curves, and lands far away. How high? How far?

Let us split the motion in two and find out.

1 · What is a projectile?

A projectile is a body thrown into the air. After the throw, only gravity acts on it.

Its path is a parabola. Think of a cricket ball or a stone.

The book splits the motion in two parts. Sideways: no acceleration, so the speed stays the same. Upward: acceleration is −g.

The starting parts are vᵢₓ = vᵢ cos θ and vᵢ_y = vᵢ sin θ.

2 · Three results

Maximum height: h = vᵢ2 sin2θ ÷ 2g.

Time of flight: t = 2vᵢ sin θ ÷ g.

Range: R = (vᵢ2 ÷ g) sin 2θ.

The range is largest when sin 2θ = 1. That is θ = 45°.

3 · Air and the book's example

With air resistance the path is skewed. It is not a perfect parabola. The range is shorter.

Book Example 2.6: vᵢ = 30 m s−1, θ = 30°.

The book gets t = 3.1 s, h = 11.5 m and R = 79.53 m.

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Notes, short questions and MCQs

Read the full notes: key terms, model answers and MCQs with answers

Projectile motion

A batter hits a ball. It rises, curves, and lands far away. How high? How far?

Let us split the motion in two and find out.

1 · What is a projectile?

A projectile is a body thrown into the air. After the throw, only gravity acts on it.

Its path is a parabola. Think of a cricket ball or a stone.

The book splits the motion in two parts. Sideways: no acceleration, so the speed stays the same. Upward: acceleration is −g.

The starting parts are vᵢₓ = vᵢ cos θ and vᵢ_y = vᵢ sin θ.

2 · Three results

Maximum height: h = vᵢ2 sin2θ ÷ 2g.

Time of flight: t = 2vᵢ sin θ ÷ g.

Range: R = (vᵢ2 ÷ g) sin 2θ.

The range is largest when sin 2θ = 1. That is θ = 45°.

3 · Air and the book's example

With air resistance the path is skewed. It is not a perfect parabola. The range is shorter.

Book Example 2.6: vᵢ = 30 m s−1, θ = 30°.

The book gets t = 3.1 s, h = 11.5 m and R = 79.53 m.

Key terms

Projectile
A body thrown into the air, moving under gravity alone.
Range
The sideways distance covered before landing.
Time of flight
Total time in the air.

Short questions with model answers

  1. Q1. What launch angle gives the largest range? Why?

      45°. Because sin 2θ = sin 90° = 1.

    Common mistakes

    • ✗ “A heavier ball travels farther.”

      ✓ The formulas have no mass in them.

    • ✗ “The sideways speed falls as the ball rises.”

      ✓ Sideways speed stays the same. Only the upward speed changes.

    MCQs

    1. 1. Which angle gives the maximum range?

      1. (a) 30°
      2. (b) 45°
      3. (c) 60°
      4. (d) 90°
      Show answer

      (b) sin 90° = 1.

    2. 2. In the book's Example 2.6 the range R is:

      1. (a) 79.53 m
      2. (b) 11.5 m
      3. (c) 3.1 m
      4. (d) 30 m
      Show answer

      (a) 11.5 is the height. 3.1 is the time in seconds.

    3. 3. The sideways acceleration of a projectile (no air) is:

      1. (a) g
      2. (b) 0
      3. (c) −g
      4. (d) 2g
      Show answer

      (b) Only gravity acts, and it is downward.

    4. 4. Time of flight is:

      1. (a) vᵢ sin θ ÷ g
      2. (b) 2vᵢ sin θ ÷ g
      3. (c) vᵢ cos θ ÷ g
      4. (d) 2vᵢ cos θ ÷ g
      Show answer

      (b) Up and down take equal time, so double.

    Quick revision

    • h = vᵢ2 sin2θ ÷ 2g, t = 2vᵢ sin θ ÷ g, R = (vᵢ2 ÷ g) sin 2θ.
    • Largest range at 45°.

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