Calculating the Average Velocity of a Particle at the Highest Point of Trajectory

Understanding the motion of a projectile is a fundamental concept in physics, and it is often analyzed through various parameters like initial velocity, angle of projection, and time. One of the key aspects to explore is the average velocity of the particle during its trajectory, specifically its highest point. This article will guide you through the process of determining the average velocity of a particle projected with an initial speed v at an angle phi;.

Step-by-Step Derivation of Average Velocity

When a particle is projected from the ground with an initial speed v at an angle phi; with the horizontal, we can break down the initial velocity into its horizontal and vertical components to analyze the projectile motion. These components are:

v_x (horizontal component): v_x v cos{phi}

v_y (vertical component): v_y v sin{phi}

The first step in calculating the average velocity is to determine the time it takes for the particle to reach the highest point of its trajectory. At the highest point, the vertical component of the velocity becomes zero. The relevant kinematic equation is:

v_y v_{y0} - g t

Setting v_y 0 at the highest point:

0 v sin{phi} - g t

Solving for t gives:

t frac{v sin{phi}}{g}

With the time determined, we can find the vertical displacement (height) h at the highest point using the following equation:

h v_{y0} t - frac{1}{2} g t^2

Substituting t into the equation:

h v sin{phi} left(frac{v sin{phi}}{g}right) - frac{1}{2} g left(frac{v sin{phi}}{g}right)^2

Simplifying the expression:

h frac{v^2 sin^2{phi}}{g} - frac{v^2 sin^2{phi}}{2g} frac{v^2 sin^2{phi}}{2g}

The horizontal displacement x at time t is given by:

x v_x t v cos{phi} left(frac{v sin{phi}}{g}right)

Further simplifying:

x frac{v^2 sin{phi} cos{phi}}{g}

The average velocity vec{V}_{avg} can be defined as the total displacement vector divided by the total time to reach the highest point. The displacement vector is xh, and the time is t. The components of the average velocity are:

V_{avgx} frac{x}{t} frac{frac{v^2 sin{phi} cos{phi}}{g}}{frac{v sin{phi}}{g}} frac{v cos{phi}}{2}

V_{avgy} frac{h}{t} frac{frac{v^2 sin^2{phi}}{2g}}{frac{v sin{phi}}{g}} frac{v sin{phi}}{2}

Therefore, the average velocity vector is:

vec{V}_{avg} left(frac{v cos{phi}}{2}, frac{v sin{phi}}{2}right)

The magnitude of the average velocity is calculated using the Pythagorean theorem:

V_{avg} sqrt{left(frac{v cos{phi}}{2}right)^2 left(frac{v sin{phi}}{2}right)^2} frac{v}{2}

Conclusion

In conclusion, the average velocity of a particle projected with an initial speed v at an angle phi; is:

vec{V}_{avg} left(frac{v cos{phi}}{2}, frac{v sin{phi}}{2}right)

and its magnitude is:

V_{avg} frac{v}{2}

Understanding the average velocity at the highest point of the trajectory is crucial for various applications in physics and engineering, making this a valuable topic for further exploration.