Mastering Motion Under Gravity for Competitive Exams
Welcome to this module on Motion Under Gravity, a fundamental concept in kinematics crucial for competitive exams like JEE. We'll explore how gravity influences the motion of objects, focusing on key principles and problem-solving techniques.
Understanding Acceleration Due to Gravity
When an object is dropped or thrown, the primary force acting on it (neglecting air resistance) is gravity. This results in a constant acceleration directed downwards. On Earth's surface, this acceleration is approximately 9.8 m/s², often denoted by 'g'.
Gravity causes a constant downward acceleration.
Objects near Earth's surface experience a consistent acceleration due to gravity, pulling them towards the center of the Earth. This acceleration is independent of the object's mass.
The acceleration due to gravity, 'g', is a vector quantity. Its magnitude is approximately 9.8 m/s² near the Earth's surface. For most competitive exam problems, we assume 'g' is constant and directed vertically downwards. This means that if an object is moving upwards, gravity slows it down, and if it's moving downwards, gravity speeds it up. The value of 'g' can vary slightly with altitude and latitude, but these variations are usually ignored in introductory physics problems.
Approximately 9.8 m/s², acting vertically downwards.
Kinematic Equations in Motion Under Gravity
The standard kinematic equations for uniformly accelerated motion are directly applicable to motion under gravity. We just need to be consistent with our sign conventions.
| Standard Equation | Motion Under Gravity (Upward Motion) | Motion Under Gravity (Downward Motion) |
|---|---|---|
| v = u + at | v = u - gt | v = u + gt |
| s = ut + 1/2 at^2 | h = ut - 1/2 gt^2 | h = ut + 1/2 gt^2 |
| v^2 = u^2 + 2as | v^2 = u^2 - 2gh | v^2 = u^2 + 2gh |
Key Convention: If we define the upward direction as positive, then acceleration due to gravity 'g' will be negative (-g). If we define the downward direction as positive, then 'g' will be positive (+g). Consistency is paramount!
Vertical Motion: Dropping and Throwing Upwards
Let's analyze two common scenarios: an object dropped from rest and an object thrown vertically upwards.
Consider an object dropped from a height 'H'. Its initial velocity (u) is 0. The distance covered is 'H', and the acceleration is 'g' downwards. Using s = ut + 1/2 at², we get H = 0t + 1/2 gt², so t = sqrt(2H/g). The final velocity v = u + gt = gt = g*sqrt(2H/g) = sqrt(2gH). Now, consider an object thrown upwards with initial velocity 'u'. At its maximum height, its final velocity 'v' is 0. Using v² = u² + 2as, we get 0² = u² + 2(-g)h_max, so h_max = u²/2g. The time to reach maximum height is found using v = u + at: 0 = u - gt_top, so t_top = u/g. The total time of flight (if it returns to the same level) is 2 * t_top = 2u/g.
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h_max = u²/2g
Projectile Motion: Horizontal and Vertical Components
Projectile motion involves an object moving under gravity after being launched with an initial velocity at an angle to the horizontal. We analyze its horizontal and vertical motions independently.
The horizontal component of velocity remains constant (assuming no air resistance) because there is no horizontal acceleration. The vertical component of velocity is affected by gravity, behaving just like an object thrown vertically upwards.
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Key formulas for projectile motion include: Time of flight (T) = 2u sin θ / g, Maximum Height (H) = u² sin² θ / 2g, and Horizontal Range (R) = u² sin 2θ / g. Understanding these relationships is vital for solving problems.
The horizontal component is u cos θ. It remains constant because there is no horizontal acceleration (assuming no air resistance).
Problem-Solving Strategies
To excel in competitive exams, practice is key. Always start by drawing a diagram, identifying knowns and unknowns, choosing a consistent sign convention, and then applying the appropriate kinematic equations.
Tip: For problems involving multiple stages of motion (e.g., an object thrown up, then falling), treat each stage separately or use the final state of one stage as the initial state of the next.
Learning Resources
A comprehensive video tutorial explaining motion under gravity with examples relevant to JEE preparation.
Explains the concepts of motion under gravity, including free fall and vertical motion, with clear explanations and diagrams.
Khan Academy's detailed explanation of projectile motion, covering its components and key equations.
Covers the essential formulas and concepts of motion under gravity, tailored for competitive exam aspirants.
A discussion forum on Physics Stack Exchange providing insights and clarifications on projectile motion concepts.
Official NCERT textbook chapter on motion in a straight line, which includes foundational concepts for motion under gravity.
The official syllabus for JEE Main, highlighting the 'Kinematics' section which covers motion under gravity.
A thorough explanation of projectile motion, including derivations and examples, from a reputable physics education resource.
Provides solved examples for motion under gravity problems, demonstrating application of formulas.
A foundational video on kinematics, setting the stage for understanding motion under gravity.