Training Physics Math Vectors and Projectile Motion
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Vectors and Projectile Motion

24 min Physics Math

Vectors extend one-dimensional kinematics into two and three dimensions by giving physical quantities both magnitude and direction. Adding and resolving vectors into components is the essential skill for analyzing projectile motion, where an object moves simultaneously in the horizontal and vertical directions under the influence of gravity alone. By treating the horizontal and vertical motions independently, you can predict the trajectory, maximum height, and range of any launched object — from a basketball to a satellite in a suborbital arc.

Vectors and Projectile Motion

In two dimensions, velocities and forces decompose into components using trigonometry.

Vector Components

$$v_x = v\cos\theta, \qquad v_y = v\sin\theta$$

$$|v| = \sqrt{v_x^2 + v_y^2}, \qquad \theta = \tan^{-1}(v_y/v_x)$$

Projectile Motion (no air resistance)

Horizontal: $x = v_0\cos\theta \cdot t$ (constant speed)

Vertical: $y = v_0\sin\theta \cdot t - \frac{1}{2}gt^2$ (free fall)

Range: $R = \frac{v_0^2 \sin 2\theta}{g}$

Example 1

A ball is launched at 40 m/s at 30° above horizontal. Find the range. ($g = 10$ m/s²)

  1. $R = \frac{40^2 \sin 60°}{10} = \frac{1600 \times 0.866}{10} = 138.6$ m.
Example 2

Find the max height for the same projectile.

  1. $v_y = v_0 \sin 30° = 20$ m/s.
  2. $h = v_y^2/(2g) = 400/20 = 20$ m.
Example 3

A force of 50 N acts at 53° to the horizontal. Find horizontal and vertical components.

  1. $F_x = 50\cos 53° \approx 50(0.6) = 30$ N.
  2. $F_y = 50\sin 53° \approx 50(0.8) = 40$ N.

Practice Problems

1. Resolve a 100 m/s velocity at 60° into components.
2. Find the range of a projectile: $v_0 = 50$ m/s, $\theta = 45°$.
3. Find the magnitude of vector $(3, 4)$.
4. A ball is launched at 20 m/s at 30°. Find the max height.
5. Find the time of flight for $v_0 = 30$ m/s at 60°.
6. At what angle is range maximized?
Show Answer Key

1. $v_x = 50$ m/s, $v_y \approx 86.6$ m/s

2. $R = 2500/10 = 250$ m

3. $\sqrt{9+16} = 5$

4. $v_y = 10$, $h = 100/20 = 5$ m

5. $T = 2v_0 \sin 60°/g = 2(30)(0.866)/10 \approx 5.2$ s

6. 45°

🎯 Projectile Motion Simulator
Horizontal component vₓ
Vertical component vᵧ
Maximum height
Time of flight
Range
Energy, Work, and Power Practice Test — Physics Math