2.5 Motion Due to Gravity

One of the most common and fundamental types of motion is the motion of objects falling freely under the influence of gravity. When you drop a ball or a stone, it falls toward the Earth. This motion is not random; it follows specific rules. Understanding motion due to gravity is essential for understanding the physics of the world around us, from falling apples to orbiting satellites.

Definition

Motion due to gravity (also called free fall) is the motion of an object under the influence of the gravitational force alone, with air resistance neglected. All freely falling objects have the same acceleration, called the acceleration due to gravity (g).

Main Idea

The main idea is that gravity causes all objects to accelerate downward at the same rate, regardless of their mass, when air resistance is negligible. Near the Earth's surface, this acceleration is approximately 9.8 m/s². This means that every second an object falls, its velocity increases by about 9.8 m/s downward. This is a remarkable and counterintuitive fact: a heavy stone and a light feather will fall at the same rate if there is no air resistance.

Core Concept

Acceleration Due to Gravity (g) The acceleration due to gravity is directed downward, toward the center of the Earth. Near the Earth's surface, its value is approximately: g = 9.8 m/s² (or 32.2 ft/s²)

• After 1 second: 9.8 m/s downward.
• After 2 seconds: 19.6 m/s downward.
• After 3 seconds: 29.4 m/s downward.

Independence from Mass The text states: "All the freely falling objects have the same acceleration called the acceleration due to gravity (g) and is independent of their masses."

This is a key concept. Whether you drop a heavy stone or a light coin, they will both accelerate at 9.8 m/s² (if air resistance is ignored).

The Effect of Air Resistance The text explains: "For large object the presence of air resistance is neglected, however if we drop a small piece of paper and coin. The coin will fall faster than a sheet of paper due to air resistance as in Figure 2.11 (a)."

However, the text continues: "However, when air is removed, as in Figure 2.11 (b), the coin and the paper will experience the same acceleration due to gravity, and both the coin and the paper will fall at same rate."

This is a crucial point. In everyday life, air resistance causes lighter objects with larger surface areas (like a sheet of paper) to fall more slowly than denser objects with smaller surface areas (like a coin). But in a vacuum (no air), all objects fall at the same rate.

• When an object moves with gravity (falling down), acceleration due to gravity is taken as positive (+g).
• When an object moves against gravity (thrown up), acceleration due to gravity is taken as negative (-g).

Detailed Explanation

Motion due to gravity is a special case of uniformly accelerated motion. The acceleration is constant (approximately 9.8 m/s² downward), so we can use the standard kinematic equations to analyze the motion.

Example from Text The text provides an example: A block of mass 2 kg is dropped from the top of a building. It strikes the ground with a speed of 78.5 m/s. How much time does it take to reach the ground? (Ignore air resistance).

• Mass m = 2 kg (not needed for the calculation).
• Initial speed v_i = 0 m/s (dropped from rest).
• Final speed v_f = 78.5 m/s.
• Acceleration due to gravity g = 9.8 m/s².

Solution: Using the definition of acceleration: g = v_f - v_iΔ t Rearranging for time: Δ t = v_f - v_ig Putting values: Δ t = 78.5 m/s - 0 m/s9.8 m/s² = 8 s So, the block will reach the ground in 8 seconds.

The Famous Leaning Tower of Pisa Experiment The text mentions Galileo: "The first scientist to measure speed as distance over time was Galileo. He dropped various objects of different masses from the leaning tower of Pisa. He found that all of them reach the ground at the same time. The acceleration of freely falling bodies is called gravitational acceleration or acceleration due to gravity."

This is a historical experiment that demonstrated that the acceleration due to gravity is independent of mass.

Intermediate Level

In physics, the motion of falling objects is a classic example of uniformly accelerated motion. The acceleration due to gravity (g ≈ 9.8 m/s²) is constant near the Earth's surface. This allows us to use the kinematic equations to solve problems involving falling objects. For example, the velocity of a falling object after time t is v = gt (if dropped from rest). The distance fallen is s = 12gt². The sign convention is important: when the object is falling downward, g is positive; when it is thrown upward, g is negative (since it opposes the motion). Understanding these principles is essential for analyzing projectile motion, which is the motion of objects thrown into the air.

Important Points

• - All objects in free fall have the same acceleration due to gravity, regardless of their mass.
• - The value of g near the Earth's surface is approximately 9.8 m/s².
• - Air resistance can cause lighter objects to fall more slowly, but in a vacuum, all objects fall at the same rate.
• - The sign of g is positive when falling (with gravity) and negative when thrown up (against gravity).

Remember

• - Acceleration due to gravity (g): 9.8 m/s² downward.
• - Independent of mass: All objects fall at the same rate in a vacuum.
• - Sign convention: +g for falling, -g for rising.
• - Air resistance: Slows down lighter objects more than heavier ones.

Common Mistakes

• - Thinking heavier objects fall faster: In a vacuum, all objects fall at the same rate. The misconception arises because of air resistance in everyday life.
• - Forgetting the sign convention: When solving problems, remember to use +g for falling objects and -g for objects thrown upward.
• - Using the mass of the object in calculations: The mass of a falling object does not affect its acceleration due to gravity. You do not need the mass to find the time of fall or the final velocity.

Quick Revision

Motion due to gravity is the motion of objects falling freely under the influence of Earth's gravity. The acceleration due to gravity is approximately 9.8 m/s² and is independent of the object's mass. This means that in the absence of air resistance, all objects fall at the same rate. The sign of the acceleration is positive when the object is falling and negative when it is moving upward against gravity.

60-Second Revision

• All objects fall at the same rate (ignoring air resistance).
• Sign convention: +g falling, -g rising.
• Galileo's experiment: All objects reach the ground at the same time.
• Air resistance: Causes lighter objects to fall slower in real life.

Frequently Asked Questions (FAQs)

Q1. What is the acceleration due to gravity? Answer: The acceleration due to gravity (g) is the acceleration experienced by an object due to the Earth's gravitational pull. Near the Earth's surface, its value is approximately 9.8 m/s² directed downward.

Q2. Does the mass of an object affect its acceleration due to gravity? Answer: No. In the absence of air resistance, all objects, regardless of their mass, experience the same acceleration due to gravity. This was demonstrated by Galileo's experiment.

Q3. Why does a feather fall slower than a stone in air? Answer: A feather falls slower than a stone in air because of air resistance. The feather has a larger surface area relative to its weight, so air resistance affects it more. In a vacuum (no air), both would fall at the same rate.

Q4. What is the sign convention for acceleration due to gravity? Answer: When an object is moving downward (with gravity), g is taken as positive (+g). When an object is moving upward (against gravity), g is taken as negative (-g).

• Q5. How can you calculate the time it takes for an object to fall?
• Answer: You can use the formula:
• Δ t = v_f - v_ig
• where v_f is the final velocity, v_i is the initial velocity, and g is the acceleration due to gravity.

• Q6. What is a freely falling body?
• Answer: A freely falling body is an object that is moving under the influence of gravity alone, with no other forces (like air resistance) acting on it.

Multiple Choice Questions (MCQs)

• Q1. The acceleration due to gravity near the Earth's surface is approximately:
• A. 9.8 m/s
• B. 9.8 m/s²
• C. 9.8 N
• D. 9.8 kg

• Correct Answer: B
• Explanation: The text states that near the Earth's surface, g is approximately 9.8 m/s².

• Q2. The acceleration due to gravity is independent of:
• A. The object's mass
• B. The object's shape
• C. The object's color
• D. All of the above

• Correct Answer: A
• Explanation: The text states that all freely falling objects have the same acceleration due to gravity, independent of their masses.

• Q3. When an object is moving against gravity (thrown upward), the acceleration due to gravity is taken as:
• A. Positive (+g)
• B. Negative (-g)
• C. Zero
• D. Infinite

• Correct Answer: B
• Explanation: The text states that when an object moves against gravity, acceleration due to gravity is taken as negative (-g).

• Q4. When an object is moving with gravity (falling downward), the acceleration due to gravity is taken as:
• A. Positive (+g)
• B. Negative (-g)
• C. Zero
• D. Infinite

• Correct Answer: A
• Explanation: The text states that when an object moves with gravity, acceleration due to gravity is taken as positive (+g).

• Q5. A block is dropped from a building and strikes the ground with a speed of 78.5 m/s. How long does it take to reach the ground?
• A. 4 s
• B. 8 s
• C. 10 s
• D. 12 s

• Correct Answer: B
• Explanation: Using the formula Δ t = 78.5 - 09.8 = 8 s, as calculated in the text.

• Q6. Why does a coin fall faster than a sheet of paper in air?
• A. Because it has more mass
• B. Because it has less air resistance
• C. Because gravity pulls it more
• D. Because it is smaller

• Correct Answer: B
• Explanation: The text states that the coin falls faster due to air resistance (the paper experiences more air resistance relative to its weight).

• Q7. In a vacuum (no air), which object falls fastest?
• A. A heavy stone
• B. A light feather
• C. A coin
• D. All fall at the same rate

Correct Answer: D Explanation: The text states that when air is removed, both the coin and the paper will experience the same acceleration due to gravity and fall at the same rate.

• Q8. The famous experiment involving dropping objects from the Leaning Tower of Pisa was conducted by:
• A. Newton
• B. Einstein
• C. Galileo
• D. Bohr

• Correct Answer: C
• Explanation: The text states that Galileo dropped various objects of different masses from the leaning tower of Pisa.

• Q9. The acceleration due to gravity is directed:
• A. Upward
• B. Downward (toward the center of the Earth)
• C. Horizontally
• D. Randomly

• Correct Answer: B
• Explanation: The text states that the acceleration due to gravity is directed downward, toward the center of the earth.

• Q10. If an object is dropped from rest, its initial velocity is:
• A. 9.8 m/s
• B. 0 m/s
• C. 78.5 m/s
• D. Infinite

• Correct Answer: B
• Explanation: An object dropped from rest has an initial velocity of 0 m/s.

Frequently Asked Questions (FAQs)

Q1. What is the difference between mass and weight in the context of falling objects? Answer: Mass is the amount of matter in an object and does not change with location. Weight is the force of gravity on an object (W = mg). In free fall, the acceleration is independent of mass, but the weight (force) depends on mass. Heavier objects have more weight, but they also have more inertia, so the acceleration is the same.

Q2. Can the acceleration due to gravity change? Answer: Yes. The value of g changes slightly with altitude (it decreases as you go higher) and with latitude (it is slightly different at the poles compared to the equator). However, for most everyday calculations, g = 9.8 m/s² is a good approximation.

• Q3. What is the velocity of a falling object after 3 seconds?
• Answer: Using v = gt, the velocity after 3 seconds is 9.8 × 3 = 29.4 m/s downward.

• Q4. What is the distance fallen by an object after 2 seconds?
• Answer: Using s = 12gt², the distance fallen is 12 × 9.8 × (2)² = 19.6 m.

Q5. Why do astronauts on the Moon fall at the same rate as on Earth (ignoring air resistance)? Answer: Actually, the acceleration due to gravity on the Moon is about 1.6 m/s², which is less than on Earth. So they fall slower. However, the principle that all objects fall at the same rate (independent of mass) holds true on the Moon as well.

Q6. What is the effect of air resistance on a falling object? Answer: Air resistance opposes the motion of the falling object. As the object speeds up, air resistance increases. Eventually, the air resistance equals the weight of the object, and the net force becomes zero. The object then falls at a constant speed called terminal velocity, and its acceleration becomes zero.

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