Vector and Trajectory Dynamics Lab
This lab focuses on the **kinematics of projectiles** and the **addition of forces (vectors)**. Use the interactive tools to calculate range, find resultant forces, and verify key vector theorems.
Mission Briefing: Key Equations & Formulas
â–¼The horizontal range is given by (neglecting air resistance): $$ R = \frac{v_0^2 \sin(2\theta)}{g} $$
- $v_0$: Initial Velocity ($\text{m/s}$)
- $\theta$: Launch Angle (degrees)
- $g$: Acceleration due to Gravity ($10 \text{ m/s}^2$)
For two vectors ($P$ and $Q$) inclined at angle $\alpha$:
$$ R = \sqrt{P^2 + Q^2 + 2PQ \cos\alpha} $$
Experiment 1: Projectile Range Optimization
Calculate the horizontal range ($R$) and verify the principle of complementary angles.
Experiment 2: Vector Resolution Challenge
Find the **Resultant Force ($R$)** given two forces ($P$ and $Q$) and the angle ($\alpha$) between them.
The horizontal range ($R$) of a projectile is the **same** for two angles of projection, $\theta$ and $(90^\circ - \theta)$. These are called complementary angles. Maximum range occurs at $45^\circ$ where the complementary angle is itself $45^\circ$.
Vector Laws and Resultant Force
The Parallelogram Law and the Triangle Law are two geometrical methods used to find the **resultant** (single vector that produces the same effect) of two or more vectors. The parallelogram method is typically used for forces originating from a single point.
A **scalar** quantity (e.g., mass, distance) is defined only by magnitude. A **vector** quantity (e.g., force, velocity) is defined by both **magnitude and direction**. Vector addition follows specific rules (like the Parallelogram Law), not simple arithmetic addition.
The equilibrant force is a single force that is **equal in magnitude** but **opposite in direction** to the resultant force ($R$). It is the force required to keep the system in equilibrium.
