What is projectile motion? Explain the factors affecting the flight of a projectile in sport.

Concept understanding — Center Of Gravity Balance

The Intuition: The One Point That Holds Everything Together

Think about balancing a pencil on your finger. You slide it back and forth until you find that one spot where it stays perfectly horizontal. That spot isn't random — it's the point where the pencil's weight on the left exactly cancels the weight on the right. If you put your finger anywhere else, the pencil tips.

Now imagine a much more complicated object: a chair, a cricket bat, or even your own body. Every tiny piece of that object is pulled downward by gravity. The center of gravity is the single point where you could pretend all that weight is concentrated. If you support the object exactly at that point, it balances perfectly — no tipping, no rotation.

Note

For most objects on Earth, the center of gravity is the same as the center of mass. The distinction matters only when gravity varies significantly across the object (e.g., a very tall building), which you won't see in school-level problems.

The Precise Statement

The center of gravity of an object is the point at which the entire weight of the object can be considered to act, for the purpose of analyzing translational and rotational equilibrium under gravity.

Mathematically, for a system of particles with weights $w_1, w_2, \dots, w_n$ located at positions $(x_1, y_1, z_1), \dots, (x_n, y_n, z_n)$ , the coordinates of the center of gravity are:

x_{CG} = \frac{\sum w_i x_i}{\sum w_i}, \quad y_{CG} = \frac{\sum w_i y_i}{\sum w_i}, \quad z_{CG} = \frac{\sum w_i z_i}{\sum w_i}

Since weight $w_i = m_i g$ , and $g$ is constant, this simplifies to the center of mass formula:

x_{CM} = \frac{\sum m_i x_i}{\sum m_i}, \quad y_{CM} = \frac{\sum m_i y_i}{\sum m_i}, \quad z_{CM} = \frac{\sum m_i z_i}{\sum m_i}

\vec{R}_{CG} = \frac{\sum m_i \vec{r}_i}{\sum m_i}

Why It Matters …

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A projectile flies freely under gravity; its flight depends on angle of projection, speed of projection, height of release, gravity and air resistance.

Projectile motion is the motion of a body that has been launched (thrown, kicked, struck or released) into the air and then moves freely under the influence of gravity alone — for example a javelin, shot, discus or a long-jumper's body. Its path is a symmetrical parabola when launch and landing heights are equal.

R = (v² sin(2θ)) / g (range for equal launch and landing height)

Factors affecting the flight:

Angle of projection (θ): on level ground the range is maximum at 45°, because sin(2θ) is then greatest.
Speed / velocity of projection (v): range is proportional to v², so doubling the release speed quadruples the range — the strongest factor. …

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Q.What is projectile motion? Explain the factors affecting the flight of a projectile in sport.

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