Deriving Projectile Formulae

Projectile Motion: This describes a particle moving freely through a two-dimensional plane under the sole influence of gravity, typically following a parabolic trajectory. The motion is defined by an initial speed $U$ launched at an angle $\theta$ relative to the horizontal plane.

Vector Components: The initial velocity must be decomposed into its horizontal ($u_x = U \cos\theta$) and vertical ($u_y = U \sin\theta$) components to analyze the motion separately. This decomposition is critical because the forces acting in each direction are independent, allowing for the use of separate kinematic equations.

Modeling Assumptions: Standard derivations assume a constant gravitational acceleration $g$, no air resistance, and that the launch and landing points are on the same vertical level. These simplifications allow for the derivation of elegant, closed-form algebraic expressions for key physical quantities.

Deriving Time of Flight (T): Set vertical displacement $s_y = 0$ in the formula $s = ut + \frac{1}{2}at^2$. Using $u_y = U \sin\theta$ and $a_y = -g$, the equation becomes $0 = (U \sin\theta)T - \frac{1}{2}gT^2$, which simplifies to $T = \frac{2U \sin\theta}{g}$.

Deriving Maximum Height (H): Use the formula $v^2 = u^2 + 2as$ for the vertical component where $v_y = 0$. By substituting $0 = (U \sin\theta)^2 - 2gH$, we isolate $H$ to get $H = \frac{U^2 \sin^2\theta}{2g}$.

Deriving Horizontal Range (R): Substitute the time of flight $T$ into the horizontal distance formula $x = u_x t$. This yields $R = (U \cos\theta) \cdot \frac{2U \sin\theta}{g} = \frac{U^2(2 \sin\theta \cos\theta)}{g}$, which simplifies via trigonometric identity to $R = \frac{U^2 \sin 2\theta}{g}$.

Displacement vs. Distance: Note that the range $R$ is a horizontal displacement. If a projectile is fired from a cliff, the range formula derived for horizontal ground does not apply because the vertical displacement $s_y$ at landing is not zero.

Angle of Projection: The range is maximized when $\sin 2\theta$ is maximized. This occurs at $\theta = 45^\circ$, whereas the maximum height is achieved by projecting vertically at $\theta = 90^\circ$.

Check the Landing Level: Always verify if the projectile lands at the same height it was launched. If the landing level is different, you cannot use the derived range formula $R = \frac{U^2 \sin 2\theta}{g}$ and must solve using basic SUVAT equations instead.

Trigonometric Precision: Be extremely careful with the difference between $\sin^2\theta$ (used in height) and $\sin 2\theta$ (used in range). Mixing these up is a frequent source of lost marks in derivation questions.

Reasonableness Checks: Ensure your derived units are correct (e.g., $U^2/g$ has units of meters). If your formula for range results in $U/g$, you have likely made an algebraic error during derivation.

Total Velocity at Apex: A common error is assuming the total velocity is zero at the maximum height. While the vertical component is zero, the horizontal component remains $U \cos\theta$, meaning the projectile is still moving at its minimum speed.

Gravity Sign Convention: Mixing positive and negative directions for $u_y$ and $g$ often leads to negative time or distance results. Always define a consistent 'upward is positive' or 'downward is positive' frame before starting the derivation.

Air Resistance Reality: In theoretical derivations, air resistance is ignored, but in real-world scenarios, it would reduce both the range and the maximum height while making the trajectory non-symmetrical.

Energy Conservation: The maximum height formula $H = \frac{U^2 \sin^2\theta}{2g}$ can also be viewed through energy. The initial vertical kinetic energy $\frac{1}{2}m(U \sin\theta)^2$ is converted entirely into gravitational potential energy $mgH$ at the peak.

Complementary Angles: For a fixed initial speed, projectiles launched at angles $\theta$ and $90^\circ - \theta$ will achieve the same horizontal range. This is because $\sin(2(90-\theta)) = \sin(180-2\theta) = \sin 2\theta$, showing the mathematical symmetry in range outcomes.