Projectile motion is a form of motion experienced by an object or particle (a projectile) that is projected near the Earth's surface and moves along a curved path under the action of gravity only. We assume air resistance is negligible.
1. Fundamental Principles
The motion is decomposed into two independent perpendicular components:
- Horizontal Component (\(x\)): Uniform velocity (acceleration \(a_x = 0\)).
- Vertical Component (\(y\)): Uniform acceleration (acceleration \(a_y = -g\)).
Initial Velocity Decomposition
If an object is launched with velocity \(u\) at an angle \(\theta\):
\[u_x = u \cos(\theta)\]
\[u_y = u \sin(\theta)\]
2. Key Derived Formulas
A. Time of Flight (\(T\))
The total time the projectile stays in the air.
\[T = \frac{2u \sin(\theta)}{g}\]
B. Maximum Height (\(H\))
The highest vertical displacement. At this point, vertical velocity \(v_y = 0\).
\[H = \frac{u^2 \sin^2(\theta)}{2g}\]
Note: Uses \(\sin^2(\theta)\) because height depends on the square of the vertical velocity component.
C. Horizontal Range (\(R\))
The total horizontal distance covered.
\[R = \frac{u^2 \sin(2\theta)}{g}\]
Note: Uses \(\sin(2\theta)\) based on the identity \(2\sin\theta\cos\theta = \sin2\theta\).
3. Important Phenomena & Rules
Complementary Angles: For a fixed velocity \(u\), the range \(R\) is identical for angles \(\theta\) and \((90^\circ - \theta)\). For example, \(30^\circ\) and \(60^\circ\) land at the same spot.
Maximum Range Angle: Range is maximum when \(\theta = 45^\circ\). At this angle, \(H = \frac{R}{4}\).
4. Testing Combinations: Explanation Guide
Use these combinations in the simulator to visualize the mathematical relationships:
1. The Shallow Launch (\(15^\circ\))
Low height and short range due to very low Time of Flight.
2. The High Lob (\(75^\circ\))
Complementary to \(15^\circ\). It reaches the same range but takes much longer to land.
3. Maximum Distance (\(45^\circ\))
Optimizes the balance between vertical "hang-time" and horizontal speed.
4. Velocity Doubling (\(u \to 2u\))
Observe that doubling velocity results in 4 times the Range (\(R \propto u^2\)).
5. The \(H=R\) Case (\(76^\circ\))
At \(\tan \theta = 4\), the height reached equals the horizontal distance covered.
6. Pure Vertical (\(90^\circ\))
Zero range, maximum possible height for a given velocity.
7. The Standard Ratio (\(30^\circ\) vs \(60^\circ\))
Check that \(60^\circ\) reaches 3 times the height of \(30^\circ\) (\(H \propto \sin^2 \theta\)).
8. Mid-Range Efficiency (\(30^\circ\))
Commonly used in problems because \(\sin 30^\circ = 0.5\).
9. Horizontal Velocity at Peak
Observe that the projectile still moves forward at the top with velocity \(u \cos \theta\).
10. Symmetry of Speed
The speed of the projectile when it hits the ground is equal to the launch speed \(u\).
5. Equation of Trajectory
The path of a projectile is a Parabola, defined by:
\[y = x \tan(\theta) - \frac{gx^2}{2u^2 \cos^2(\theta)}\]
