Important Derivations: Motion in a Plane | Physics for JEE Main & Advanced PDF Download

Derivation 1

Time of Flight, Horizontal Range, Maximum Height and Equation of Path of Projectile in Projectile Motion

Consider a projectile launched with an initial velocity v₀ at an angle θ with respect to the horizontal axis. We’ll assume no air resistance and a constant acceleration due to gravity g. The horizontal and vertical components of velocity (v_x and v_y) can be calculated as follows:

The time of flight (T) is the total time the projectile stays in the air until it returns to the same vertical position from which it was launched. The time of flight can be found using the vertical motion equation:

Solving for T:

Now, the horizontal distance (R) covered by the projectile can be calculated using the time of flight:

Using the identity sin(2θ)=2sin(θ)cos(θ), we can simplify:

Now, let’s find the maximum height (H) reached by the projectile. The maximum height occurs when the vertical component of velocity becomes zero at the top of the trajectory. We can use the vertical motion equation:

Solving for t_top:

Now, we can find the maximum height (H) using the vertical motion equation:


Finally, the equation of the path of the projectile (y as a function of x) is a parabolic curve:

This equation represents the trajectory of the projectile launched at an angle θ from the point (x₀,y₀), neglecting air resistance.

Important Formulas Related to Projectile Motion

1. Horizontal Velocity (vx) of the Projectile:
   
2. Vertical Velocity (vy) of the Projectile:
   
3. Time of Flight (T):
   
4. Maximum Height (H):
   
5. Range (R):
   
6. Vertical Displacement (y) at Time t:
   
7. Horizontal Displacement (x) at Time t:
   
8. Equation of the Projectile’s Path:
   
   where: v₀ = Initial velocity of the projectile
   θ = Launch angle
   g = Acceleration due to gravity (approximately 9.81 m/s² on Earth)