Projectile Motion Notes
Equations of Motion
- Position as a function of time: \[ r = r_0 + v_0 t + \frac{1}{2} a t^2 \]
- Break into x and y components: \[ x = x_0 + v_{0x} t + \frac{1}{2} a_x t^2 \] \[ y = y_0 + v_{0y} t + \frac{1}{2} a_y t^2 \]
- Horizontal and vertical motions are independent!
Projectile Motion Basics
- Horizontal motion: No acceleration (\(a_x = 0\)). Speed remains constant: \[ v_{0x} = v_o \cos \theta_o \]
- Vertical motion: Acceleration due to gravity (\(a_y = -g\)): \[ v_{0y} = v_o \sin \theta_o \]
- Path is a parabola: \[ y = x \tan \theta_o – \frac{g x^2}{2 (v_o \cos \theta_o)^2} \]
Key Formulas
- Time to reach max height: \[ t_m = \frac{v_o \sin \theta_o}{g} \]
- Total time of flight: \[ T_f = \frac{2 v_o \sin \theta_o}{g} \]
- Max height: \[ h_m = \frac{(v_o \sin \theta_o)^2}{2g} \]
- Horizontal range: \[ R = \frac{v_o^2 \sin 2\theta_o}{g} \] Max range at \(\theta_o = 45^\circ\): \(R_{max} = \frac{v_o^2}{g}\)
Important Concepts for NEET
- Independence of motions: Horizontal (constant speed) and vertical (constant acceleration) motions are separate.
- Key equations: Time of flight (\(T_f\)), max height (\(h_m\)), and range (\(R\)) are frequently tested.
- Symmetry: Time to max height is half the total flight time (\(t_m = T_f / 2\)).
- Range at angles: Angles \(45^\circ + \alpha\) and \(45^\circ – \alpha\) give the same range.
- Parabolic path: Projectiles follow a parabola due to constant horizontal velocity and vertical acceleration.
Example Summaries
- Stone thrown horizontally:
- Time to fall: \( t = \sqrt{\frac{2h}{g}} = 10 \, \text{s} \) (for \( h = 490 \, \text{m} \))
- Final speed: \( \sqrt{v_x^2 + v_y^2} = 99 \, \text{m/s} \)
- Cricket ball at \(30^\circ\):
- Max height: \(10.0 \, \text{m}\)
- Time of flight: \(2.9 \, \text{s}\)
- Range: \(69 \, \text{m}\)
