Horizontal & Vertical Components (AQA A-Level Mathematics): Revision Notes

• Horizontal Component: This is the part of the vector that runs parallel to the horizontal axis (often the $x$ -axis).
• Vertical Component: This is the part of the vector that runs parallel to the vertical axis (often the $y$ -axis).

• Horizontal component $v_x$ : $v_x = v \cos \theta$
• Vertical component $v_y$ : $v_y = v \sin \theta$

• Horizontal Motion:
• The horizontal component $v_x = v \cos \theta$ remains constant if air resistance is neglected because there is no horizontal acceleration (assuming a flat surface).
• Displacement in the horizontal direction is given by $x = v_x \cdot t$ , where $t$ is the time of flight.

• Vertical Motion:
• The vertical component $v_y = v \sin \theta$ is affected by gravity.
• The vertical motion is typically analysed using the equations of motion under constant acceleration $g$ (9.81 m/s² downward).

Problem: A ball is thrown with an initial velocity of 20 m/s at an angle of 30° above the horizontal. Question : Find the horizontal and vertical components of the velocity.

Step 1: Identify the given values:

• Initial velocity $v = 20 \, \text{m/s}$
• Angle $\theta = 30^\circ$

Step 2: Calculate the horizontal and vertical components:

• Horizontal component: $v_x = 20 \cos 30^\circ = 20 \times \frac{\sqrt{3}}{2} \approx 17.32 \, \text{m/s}$

• Vertical component: $v_y = 20 \sin 30^\circ = 20 \times \frac{1}{2} = 10 \, \text{m/s}$ Answer:

• The horizontal component of the velocity is 17.32 m/s.

• The vertical component of the velocity is 10 m/s.

• Independence of motion: Horizontal and vertical components are independent of each other. The horizontal motion doesn't affect the vertical motion and vice versa, except they are linked by the time of flight.
• Simplifying calculations: Breaking vectors into components simplifies the analysis of motion, particularly in two dimensions.