Understanding Motion Under Gravity: Complete Guide for Students

Motion under gravity refers to the vertical motion of objects when only the force of gravity is acting on them, without any influence from air resistance or other external forces. This topic introduces key equations, concepts, and problem-solving approaches essential for JEE Main and similar examinations.

Physical Concept of Motion Under Gravity

When an object moves vertically under the force of gravity alone, it undergoes constant acceleration directed towards the centre of the Earth. The magnitude of this acceleration is called the acceleration due to gravity, denoted by $g$, with a standard value of approximately $9.8\,\mathrm{m/s^2}$ near the Earth's surface.

This type of motion is a specific instance of one-dimensional motion with uniform acceleration, where gravity is the only acting force. The direction of acceleration is always downward, regardless of the object's initial velocity or direction of motion. For additional detail on uniform motion, refer to the One-Dimensional Motion Overview.

Standard Equations for Vertical Motion Under Gravity

The equations of motion under gravity are specialized forms of the kinematic equations for constant acceleration, with $a$ replaced by $g$. The sign convention must be defined at the outset: generally, upward is taken as positive, so acceleration due to gravity is negative ($a = -g$).

For motion under gravity, the following symbols are used:

• $u$: Initial velocity (upward or downward)
• $v$: Final velocity after time $t$
• $t$: Time elapsed
• $s$: Vertical displacement
• $g$: Acceleration due to gravity ($9.8\,\mathrm{m/s^2}$)

Key equations are as follows:

• $v = u + gt$
• $s = ut + \dfrac{1}{2}gt^2$
• $v^2 = u^2 + 2gs$

Application of these equations requires careful attention to the direction and sign of each quantity. For advanced understanding of gravitational effects, refer to Understanding Gravity's Acceleration.

Types of Vertical Motion: Free Fall and Upward Projection

Motion under gravity can be classified mainly into two types: free fall and upward throw. In free fall, the object is released from rest and accelerates downwards. In upward projection, the object is projected vertically upwards with an initial velocity and eventually returns downward under gravity.

For free fall, velocity increases in the downward direction. For upward projection, the velocity decreases until it becomes zero at the highest point, after which the object undergoes downward motion as a free-falling body. Related concepts can be further explored in Kinematics Principles.

Derivation of Motion Under Gravity Equations

The standard kinematic equations are modified for gravity by substituting acceleration $a$ with $-g$ (if upward is positive) or $+g$ (if downward is positive). It is essential to state the sign convention before solving problems to avoid confusion.

If an object is thrown upward ($u > 0$), the velocity after time $t$ is $v = u - gt$ (using upward as positive). The displacement after time $t$ can be written as $s = ut - \dfrac{1}{2}gt^2$. For an object dropped from rest, $u = 0$, so $v = gt$ and $s = \dfrac{1}{2}gt^2$.

Worked Example: Motion Under Gravity Applications

A stone is dropped from a height of $40\,\mathrm{m}$ ($u = 0$) and $g = 10\,\mathrm{m/s^2}$. The time to reach the ground is found using $s = \dfrac{1}{2}gt^2$. Setting $40 = \dfrac{1}{2} \times 10 \times t^2$, we obtain $t^2 = 8$ or $t = 2.83\,\mathrm{s}$.

The final velocity just before hitting the ground is $v = gt = 10 \times 2.83 = 28.3\,\mathrm{m/s}$. Such calculations frequently appear in examination questions and require the correct sign convention for $g$. For additional example problems and deeper discussion, refer to Projectile Motion Concepts.

Significance of the Acceleration Due to Gravity

The value of $g$ varies marginally with altitude and latitude, but for standard problems it is taken as $9.8\,\mathrm{m/s^2}$ or rounded to $10\,\mathrm{m/s^2}$ for simplicity. Acceleration due to gravity does not depend on the mass of the object.

All objects, if air resistance is ignored, fall toward the Earth with the same acceleration $g$, regardless of their properties. This fundamental principle is a basis for classical mechanics.

Common Applications and Problem Types

Typical applications of the motion under gravity equations include calculating time of flight, determining maximum height in upward throws, and evaluating velocity at particular positions. These concepts form a foundation for more advanced topics such as Rotational Motion Basics.

• Finding the time of flight for vertical projectile motion
• Calculating maximum height reached by an object
• Determining final velocity before impact

Key Points and Common Mistakes

Solving numerical problems in motion under gravity requires strict attention to direction, initial conditions, and sign conventions for all quantities. Incorrect handling of signs or confusion between initial and final velocities are common sources of error.

• Incorrect sign convention for $g$
• Mixing initial and final velocities
• Overlooking air resistance in specific situations
• Confusing equations for upward and downward motion

Regular practice and careful consideration of problem statements minimize errors and improve accuracy in examinations. For concept revision, comparison with other mechanical concepts such as Work vs Energy Explained is recommended.