Projectile Motion Formulas and Trajectory Equations

Fundamental Displacements in Projectile Motion

• Horizontal Displacement after $t$ Seconds:

  • Formula:     $x = (u \cos(\theta))t$

  • Parameters:

  • $x$: Horizontal displacement attained after time $t$

  • $u$: Initial velocity of launch

  • $\theta$: Angle of projection relative to the horizontal axis

  • $t$: Time elapsed in seconds

  • Physical Mechanism: Horizontal motion occurs at a constant velocity because there is no horizontal acceleration ($a_x = 0$). The effective horizontal component of initial velocity is $u_x = u \cos(\theta)$.

• Vertical Displacement after $t$ Seconds:

  • Formula:     $y = (u \sin(\theta))t - \frac{1}{2} g t^2$

  • Parameters:

  • $y$: Vertical displacement attained after time $t$

  • $u$: Initial velocity of launch

  • $\theta$: Angle of projection relative to the horizontal axis

  • $g$: Acceleration due to gravity

  • $t$: Time elapsed in seconds

  • Physical Mechanism: Vertical motion is uniformly accelerated due to gravitational attraction acting downwards ($a_y = -g$). The initial vertical velocity component is $u_y = u \sin(\theta)$.

Trajectory Equation of Projectile Motion

• Equation of the Path of Projectile:

  • Formula:     $y = x \tan(\theta) - \frac{g}{2u^2 \cos^2(\theta)} x^2$

  • Parameters:

  • $y$: Vertical position coordinate of the projectile at horizontal distance $x$

  • $x$: Horizontal position coordinate of the projectile

  • $\theta$: Angle of projection relative to the horizontal

  • $u$: Initial projection speed

  • $g$: Acceleration due to gravity

  • Derivation and Mathematical Characteristics:

  • Rearranging the horizontal displacement equation for time yields $t = \frac{x}{u \cos(\theta)}$.

  • Substituting this time expression into the vertical displacement equation gives:       $y = u \sin(\theta) \left(\frac{x}{u \cos(\theta)}\right) - \frac{1}{2} g \left(\frac{x}{u \cos(\theta)}\right)^2$

  • Simplifying yields the trajectory equation: $y = x \tan(\theta) - \frac{g}{2u^2 \cos^2(\theta)} x^2$.

  • Because $y$ is expressed as a quadratic function of $x$ with a negative leading coefficient, the path followed by any projectile in a uniform gravitational field is parabolic.

Key Trajectory Quantities

• Time of Flight ($T$):

  • Formula:     $T = \frac{2u \sin(\theta)}{g}$

  • Parameters:

  • $T$: Total time the projectile remains in flight before returning to the original launch level

  • $u$: Initial projection speed

  • $\theta$: Angle of projection

  • $g$: Acceleration due to gravity

  • Details: Calculated by setting total vertical displacement $y = 0$ in the displacement formula, giving $t(u \sin(\theta) - \frac{1}{2}gt) = 0$, which yields non-zero solution $T = \frac{2u \sin(\theta)}{g}$.

• Maximum Height of Projectile ($H$):

  • Formula:     $H = \frac{u^2 \sin^2(\theta)}{2g}$

  • Parameters:

  • $H$: Peak vertical position reached by the projectile above the launch point

  • $u$: Initial projection speed

  • $\theta$: Angle of projection

  • $g$: Acceleration due to gravity

  • Details: Occurs at time $t_{\text{peak}} = \frac{T}{2} = \frac{u \sin(\theta)}{g}$, where the vertical velocity component vanishes ($v_y = 0$).

• Horizontal Range of Projectile ($R$):

  • Formula:     $R = \frac{u^2 \sin(2\theta)}{g}$

  • Parameters:

  • $R$: Total horizontal distance traveled by the projectile from launch to landing

  • $u$: Initial projection speed

  • $\theta$: Launch angle relative to horizontal

  • $g$: Acceleration due to gravity

  • Details: Derived by evaluating horizontal displacement $x$ at total time of flight $T$:     $R = (u \cos(\theta)) \times \left(\frac{2u \sin(\theta)}{g}\right) = \frac{u^2 (2 \sin(\theta) \cos(\theta))}{g} = \frac{u^2 \sin(2\theta)}{g}$

• Maximum Horizontal Range ($R_{\text{max}}$):

  • Formula:     $R_{\text{max}} = \frac{u^2}{g}$

  • Condition for Maximum Range:     $\theta = 45^\circ$

  • Mathematical Explanation: The sine function $\sin(2\theta)$ attains its maximum possible value of $1$ when its argument is $90^\circ$ ($2\theta = 90^\circ$), corresponding to an optimal launch angle of $\theta = 45^\circ$.

Summary of Projectile Formulas

• Horizontal range of projectile:   $R = \frac{u^2 \sin(2\theta)}{g}$

• Time of flight:   $T = \frac{2u \sin(\theta)}{g}$

• Maximum height of Projectile:   $H = \frac{u^2 \sin^2(\theta)}{2g}$

• Maximum horizontal range ($\theta = 45^\circ$):   $R_{\text{max}} = \frac{u^2}{g}$

• Equation of the path of projectile:   $y = x \tan(\theta) - \frac{g}{2u^2 \cos^2(\theta)} x^2$

• Vertical displacement after $t$ seconds:   $y = (u \sin(\theta))t - \frac{1}{2} g t^2$

• Horizontal displacement after $t$ seconds:   $x = (u \cos(\theta))t$

s = d/t (speed equals distance over time)