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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 (vx and vy) can be calculated as follows:

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

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:

Important Derivations: Motion in a Plane | Physics for JEE Main & Advanced
Solving for T:
Important Derivations: Motion in a Plane | Physics for JEE Main & Advanced
Now, the horizontal distance (R) covered by the projectile can be calculated using the time of flight:
Important Derivations: Motion in a Plane | Physics for JEE Main & Advanced
Using the identity sin(2θ)=2sin(θ)cos(θ), we can simplify:
Important Derivations: Motion in a Plane | Physics for JEE Main & Advanced
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:
Important Derivations: Motion in a Plane | Physics for JEE Main & Advanced
Solving for ttop:
Important Derivations: Motion in a Plane | Physics for JEE Main & Advanced
Now, we can find the maximum height (H) using the vertical motion equation:
Important Derivations: Motion in a Plane | Physics for JEE Main & Advanced
Important Derivations: Motion in a Plane | Physics for JEE Main & Advanced
Finally, the equation of the path of the projectile (y as a function of x) is a parabolic curve:
Important Derivations: Motion in a Plane | Physics for JEE Main & Advanced
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:
    Important Derivations: Motion in a Plane | Physics for JEE Main & Advanced
  2. Vertical Velocity (vy) of the Projectile:
    Important Derivations: Motion in a Plane | Physics for JEE Main & Advanced
  3. Time of Flight (T):
    Important Derivations: Motion in a Plane | Physics for JEE Main & Advanced
  4. Maximum Height (H):
    Important Derivations: Motion in a Plane | Physics for JEE Main & Advanced
  5. Range (R):
    Important Derivations: Motion in a Plane | Physics for JEE Main & Advanced
  6. Vertical Displacement (y) at Time t:
    Important Derivations: Motion in a Plane | Physics for JEE Main & Advanced
  7. Horizontal Displacement (x) at Time t:
    Important Derivations: Motion in a Plane | Physics for JEE Main & Advanced
  8. Equation of the Projectile’s Path:
    Important Derivations: Motion in a Plane | Physics for JEE Main & Advanced
    where: v0 = Initial velocity of the projectile
    θ = Launch angle
    g = Acceleration due to gravity (approximately 9.81 m/s² on Earth)
The document Important Derivations: Motion in a Plane | Physics for JEE Main & Advanced is a part of the JEE Course Physics for JEE Main & Advanced.
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FAQs on Important Derivations: Motion in a Plane - Physics for JEE Main & Advanced

1. What are the important derivations for motion in a straight line in JEE?
Ans. Some important derivations for motion in a straight line in JEE include the derivation of equations of motion, the derivation of the formula for velocity, the derivation of the formula for acceleration, the derivation of the formula for displacement, and the derivation of the formula for time.
2. How do you derive the equations of motion for motion in a straight line?
Ans. The equations of motion for motion in a straight line can be derived using the basic principles of kinematics. By analyzing the relationship between displacement, velocity, acceleration, and time, we can derive the equations of motion. The derivations involve mathematical equations and algebraic manipulation to derive the final formulas.
3. Can you explain the derivation of the formula for velocity in motion in a straight line?
Ans. The formula for velocity in motion in a straight line can be derived using the definition of velocity as the rate of change of displacement with respect to time. By taking the derivative of the displacement equation, we can obtain the formula for velocity. The derivation involves differentiating the displacement equation with respect to time and simplifying the resulting expression.
4. How is the formula for acceleration derived in motion in a straight line?
Ans. The formula for acceleration in motion in a straight line can be derived using the definition of acceleration as the rate of change of velocity with respect to time. By taking the derivative of the velocity equation, we can obtain the formula for acceleration. The derivation involves differentiating the velocity equation with respect to time and simplifying the resulting expression.
5. What is the derivation of the formula for displacement in motion in a straight line?
Ans. The formula for displacement in motion in a straight line can be derived by integrating the velocity equation with respect to time. By integrating the velocity equation, we can obtain the formula for displacement. The derivation involves integrating the velocity equation with respect to time and applying the appropriate limits to find the displacement.
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