Understanding Maximum Height in Projectile Motion

When a particle is projected into the air with an initial velocity at an angle to the horizontal, it follows a parabolic path. This motion is called projectile motion. One important characteristic of projectile motion is the maximum height it reaches before starting to descend. This maximum height depends on the initial velocity and the angle of projection.

Let's consider a particle projected with initial velocity $u$ at an angle $\theta$ with the horizontal. We can analyze the motion by splitting the initial velocity into its horizontal and vertical components:

• Horizontal component of initial velocity: $u_x = u \cos\theta$
• Vertical component of initial velocity: $u_y = u \sin\theta$

The motion in the vertical direction is influenced by gravity, which acts downwards (we usually take it as negative acceleration, $-g$). The maximum height is reached when the vertical component of the velocity becomes zero. At the highest point, the particle momentarily stops moving upwards before falling back down.

Deriving the Formula for Maximum Height

To find the maximum height (let's call it $H$), we can use the kinematic equation that relates initial velocity, final velocity, acceleration, and displacement in the vertical direction. The equation is:

$$ v_y^2 = u_y^2 + 2 a_y \Delta y $$

Here:

• $v_y$ is the final vertical velocity (which is 0 at maximum height).
• $u_y$ is the initial vertical velocity ($u \sin\theta$).
• $a_y$ is the vertical acceleration due to gravity ($-g$).
• $\Delta y$ is the vertical displacement (which is the maximum height $H$).

Substituting these values into the equation:

$$ 0^2 = (u \sin\theta)^2 + 2 (-g) H $$

Simplifying the equation:

$$ 0 = u^2 \sin^2\theta - 2gH $$

Now, we can rearrange the equation to solve for $H$, the maximum height:

$$ 2gH = u^2 \sin^2\theta $$

$$ H = \frac{u^2 \sin^2\theta}{2g} $$

This formula gives the maximum height attained by the projectile. It shows that the maximum height depends on the square of the initial velocity $u$, the square of the sine of the angle of projection $\theta$, and is inversely proportional to the acceleration due to gravity $g$.

Analyzing the Options for Maximum Height

Let's compare our derived formula for the maximum height with the given options:

The derived formula $H = \frac{u^2 \sin^2\theta}{2g}$ matches the formula presented in Option 3. This confirms that Option 3 correctly represents the maximum height attained by the projectile in projectile motion.

Understanding the concept of projectile motion and how to use kinematic equations is crucial for solving such problems related to maximum height, range, and time of flight.