Force and Motion 8

1. Definition of Force

Force is a physical quantity that causes a change in the state of motion or the shape of an object. It is a vector quantity, meaning it has both magnitude and direction.

Basic Definition: Force is a push or a pull that can change the state of motion or shape of an object.

Physics Perspective: Force is any interaction that, when unopposed, will change the motion of an object. It has both magnitude and direction, and is measured in Newtons (N).

Mechanical Definition: Force is a physical quantity that causes an object to accelerate, slow down, or change direction when applied.

Everyday Understanding: Force is what you use when you push a door to open it, pull a cart, or kick a ball.

Force in Nature: Force is the influence that causes objects to move or stop moving, like the force of gravity pulling things toward the ground or the force of magnetism attracting iron objects.

➡️ A force can:

• Cause a stationary object to move.
• Change the speed or direction of a moving object.
• Deform or change the shape of an object.

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2. SI Unit of Force

The SI unit of force is the Newton (N).

• 1 Newton (N) = The amount of force required to accelerate a 1 kg mass by 1 m/s².

$$1 \, \text{N} = 1 \, \text{kg} \cdot \text{m/s}^2$$

The CGS (centimeter-gram-second) system is a metric system of units that defines the unit of force as the dyne.

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3. Dimensional Formula of Force

The dimensional formula of force is derived from its definition:

$$F = m \cdot a$$

where:

• $m$ = mass → [M]
• $a$ = acceleration → [L T^{-2}]

$$\text{Dimensional formula of Force} = [M L T^{-2}]$$

4. Formula of Force

The general formula of force is based on Newton's Second Law of Motion:

$$F = m \cdot a$$

where:

• $F$ = Force
• $m$ = Mass of the object
• $a$ = Acceleration

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5. Formula derivation;

Newton’s Second Law:

According to Newton's second law, the rate of change of momentum of a body is directly proportional to the applied force and acts in the direction of the force.

$$F \propto \frac{dP}{dt}$$

where $P$ = momentum = $m \cdot v$

Taking derivative:

$$F = m \frac{dv}{dt}$$

Since $\frac{dv}{dt} = a$:

$$F = m \cdot a$$

or,

The equation $F = ma$ represents Newton's second law of motion, which states that the force acting on an object is equal to the mass of the object multiplied by its acceleration. Here's how it's derived:

1. Starting with the definition of acceleration:

Acceleration is defined as the rate of change of velocity with respect to time:

$$a = \frac{dv}{dt}$$

where:

• $a$ is acceleration,
• $v$ is velocity,
• $t$ is time.

2. Velocity as a function of time:

Now, if we rearrange the equation to express velocity change over time, we have:

$$dv = a \, dt$$

3. Relation between momentum and force:

Momentum ($p$) is defined as the product of mass ($m$) and velocity ($v$):

$$p = mv$$

The rate of change of momentum (i.e., how momentum changes with time) is what we call force:

$$F = \frac{dp}{dt}$$

4. Differentiating momentum with respect to time:

Applying the chain rule to the equation $p = mv$, where mass $m$ is constant, we get:

$$\frac{dp}{dt} = m \frac{dv}{dt}$$

Since $\frac{dv}{dt} = a$, the equation becomes:

$$F = ma$$

Conclusion:

Thus, the force $F$ is equal to the mass $m$ of an object multiplied by its acceleration $a$, which is the expression of Newton's second law of motion:

$$F = ma$$

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6. Example of Force

Example 1:

• A force of 10 N is applied to a body of mass 2 kg. Find the acceleration.

$$F = m \cdot a$$ $$10 = 2 \cdot a$$ $$a = \frac{10}{2} = 5 \, m/s^2$$

Example 2:

• If a body of mass 5 kg is accelerating at 3 m/s², the force acting on it is:

$$F = 5 \times 3 = 15 \, N$$

✅ Summary:

Property	Description
Definition	Force is an external cause that changes the state of motion or shape of an object.
SI Unit	Newton (N)
Dimensional Formula	$[M L T^{-2}]$
Formula	$F = m \cdot a$
Example	If $m = 2 \, kg$ and $a = 5 \, m/s^2$, then $F = 10 \, N$
Types	Contact and Non-contact forces

Distance and Displacement

Distance and displacement are two fundamental concepts in physics that describe the motion of an object. While they may seem similar, they are distinctly different in terms of their definitions and properties.

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1. Distance

• Definition: Distance is the total path length traveled by an object, regardless of the direction.
• Scalar Quantity: Distance is a scalar quantity, meaning it only has magnitude (size) and no direction.
• Units: It is typically measured in meters (m), kilometers (km), etc.
• Characteristics:
• It is always positive.
• It depends on the actual path taken, so the value is always equal to or greater than the displacement.
• Distance is not concerned with the starting or ending points, but with how far the object has traveled.

Distance is the total path length traveled by an object, regardless of direction.

Formula:

$$\text{Distance} = \text{Sum of the total path lengths}$$

• Unit: Meter (m)
• Note: Distance is always a positive quantity and does not consider direction.

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2. Displacement

• Definition: Displacement is the shortest straight-line distance from the initial position to the final position of an object, along with the direction of motion.
• Vector Quantity: Displacement is a vector quantity, meaning it has both magnitude (size) and direction.
• Units: Measured in meters (m), similar to distance.
• Characteristics:
• Can be positive, negative, or zero, depending on the direction.
• The displacement depends on the initial and final positions, not on the path taken.
• Displacement is the straight-line distance, and it’s the shortest path between two points.

$$\text{Displacement}=\text{Final Position}-\text{Initial Position}$$

• Unit: Meter (m)
• Note: Displacement is a vector quantity, meaning it has both magnitude and direction.

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Key Differences:

Property	Distance	Displacement
Type	Scalar quantity	Vector quantity
Direction	Does not consider direction	Has both magnitude and direction
Magnitude	Always positive and can be large	Can be positive, negative, or zero
Path Taken	Dependent on the path taken	Independent of the path taken
Example	If an object moves 10 meters in a circle, the distance is 10 meters	If an object moves 10 meters in a circle, the displacement is 0 meters (same start and end point)

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Diagram:

Let's consider an object moving along a path:

1. Initial Position (A): Start point.
2. Final Position (B): End point.

If the object moves in a curved path from point A to B, the distance is the total length of the path, while the displacement is the straight-line distance between point A and point B.

Simple Diagram:

          B
         /|
        / |
       /  | Displacement (d)
      /   |
     /    |
 A -------|
    Distance (D)

• A to B: The curved path represents the distance traveled.
• A to B (Straight line): The straight line represents the displacement.

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Example:

• If an object moves 5 meters east and then 5 meters west:
  • Distance = 5 + 5 = 10 meters (total path traveled).
  • Displacement = 0 meters (since the object ends up at the same position it started).

Here is the diagram showing the difference between distance and displacement. The curved path represents the distance, while the straight line between the points A and B represents the displacement. The direction of movement is indicated with arrows.

Speed:

• Definition: Speed is the rate at which an object covers distance. It is a scalar quantity, meaning it only has magnitude and no direction.
• Formula: $$\text{Speed} = \frac{\text{Distance}}{\text{Time}}$$
• SI Unit: Meter per second (m/s)
• Fact: Speed can be constant (uniform speed) or variable (non-uniform speed). For example, when a car travels at a constant rate of 60 km/h, it has uniform speed.

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Velocity:

• Definition: Velocity is the rate at which an object changes its position. It is a vector quantity, meaning it has both magnitude and direction.
• Formula: $$\text{Velocity} = \frac{\text{Displacement}}{\text{Time}}$$
• SI Unit: Meter per second (m/s)
• Fact: The direction of velocity is important. For example, an object moving at 60 km/h eastward has a different velocity than one moving at the same speed westward.

✅ Detailed Explanation of Scalar and Vector Quantities

🔹 Scalar Quantity

A scalar quantity is fully described by magnitude (size or value) only. It does not have a direction. Scalars are represented by numerical values along with their units.

📌 Key Characteristics of Scalars:

• Only magnitude is required to describe them.
• No direction involved.
• Scalars follow ordinary arithmetic for addition and subtraction.

🚀 Examples of Scalar Quantities:

Quantity	Definition	Example	Unit
Speed	Rate of motion without direction	60 km/h	m/s, km/h
Distance	Total path covered by an object	5 meters	m, km
Time	Duration of an event	2 hours	s, min, hr
Mass	Quantity of matter in a body	10 kg	kg, g
Temperature	Degree of hotness or coldness	30°C	°C, K

🔎 Example of Scalar Calculation:

If you drive a car at 60 km/h for 2 hours, the total distance covered is:

$$\text{Distance} = \text{Speed} \times \text{Time} = 60 \times 2 = 120 \, \text{km}$$

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🔹 Vector Quantity

A vector quantity requires both magnitude and direction to be fully defined. It is represented by an arrow, where:

• Length of the arrow = Magnitude
• Direction of the arrow = Direction of the quantity

📌 Key Characteristics of Vectors:

• Defined by both magnitude and direction.
• Follows vector addition rules (triangle law or parallelogram law).
• Can be represented in terms of components (x, y, z).

🚀 Examples of Vector Quantities:

Quantity	Definition	Example	Unit
Velocity	Rate of change of displacement with direction	60 km/h north	m/s, km/h
Displacement	Shortest distance between two points with direction	5 meters east	m, km
Force	Push or pull with direction	10 N at 30°	N (Newton)
Acceleration	Rate of change of velocity with direction	9.8 m/s² downward	m/s²
Momentum	Mass × velocity

Scalar and Vector Quantities

Type Definition Example Scalar Quantity A scalar quantity is defined by only its magnitude (size or value) and has no direction. - Speed (e.g., 60 km/h) - Distance (e.g., 5 meters) - Time (e.g., 2 hours) - Mass (e.g., 10 kg) Vector Quantity A vector quantity is defined by both magnitude and direction. - Velocity (e.g., 60 km/h north) - Displacement (e.g., 5 meters east) - Force (e.g., 10 N at 30°) - Acceleration (e.g., 9.8 m/s² downward)

Differences Between Scalar and Vector Quantities

Property Scalar Vector Definition Only magnitude Magnitude and direction Representation Single value Arrow (length = magnitude, direction = direction) Addition Rule Simple arithmetic addition Follows the vector addition rule (triangle/parallelogram law) Example Speed = 50 km/h Velocity = 50 km/h north

Diagram Explanation

Scalar Quantity: A scalar quantity is represented as a simple value. Example: If you walk 5 meters, the distance is 5 meters (no direction involved). Vector Quantity: A vector quantity is represented as an arrow, where: Length = Magnitude Arrowhead = Direction Example: If you walk 5 meters towards the east, it becomes a displacement vector.

🚀 Speed vs Velocity

Term             Definition Speed The rate at which an object covers distance. It is a scalar quantity that has only magnitude (size) but no direction. Velocity The rate at which an object changes its position in a specific direction. It is a vector quantity that has both magnitude and direction.

🔎 About Speed

Speed measures how fast an object is moving, irrespective of the direction. It only considers the distance traveled, not the displacement. Since speed has no direction, it is a scalar quantity.

✅ Examples of Speed:

A car moving at 60 km/h on a highway. A person running at 5 m/s on a track. A train traveling at 100 km/h.

📏 Formula for Speed:

$$\text{Speed} = \frac{\text{Distance}}{\text{Time}}$$

🌍 SI Unit of Speed:

Meters per second (m/s) Other units: km/h, miles/hour (mph)

🔎 Example Calculation:

If a car travels 150 km in 3 hours, the speed is: $$\text{Speed} = \frac{150 \, \text{km}}{3 \, \text{hours}} = 50 \, \text{km/h}$$

🔎 About Velocity

Velocity measures how fast an object is moving AND in which direction. It considers the displacement (shortest path) rather than distance. Velocity is a vector quantity.

✅ Examples of Velocity:

A car moving at 60 km/h north. A plane flying at 800 km/h west. A ball rolling at 5 m/s down a slope.

📏 Formula for Velocity:

$$\text{Velocity} = \frac{\text{Displacement}}{\text{Time}}$$

🌍 SI Unit of Velocity:

Meters per second (m/s) Other units: km/h, miles/hour (mph)

🔎 Example Calculation:

If a person moves 120 m east in 20 seconds, the velocity is: $$\text{Velocity} = \frac{120 \, m}{20 \, s} = 6 \, m/s \, \text{east}$$

🔥 Difference Between Speed and Velocity

Property Speed Velocity Definition Rate of change of distance with time Rate of change of displacement with time Type Scalar (no direction) Vector (magnitude and direction) Formula $\text{Speed} = \frac{\text{Distance}}{\text{Time}}$ $\text{Velocity} = \frac{\text{Displacement}}{\text{Time}}$ Direction No Yes Magnitude Always positive Can be positive or negative (depending on direction) Example Car traveling at 60 km/h Car traveling at 60 km/h north

🔎 Graphical Representation

✅ Speed: If a car moves 60 km in 1 hour, the speed is: bash |----|----|----|----|----> 60 km/h ✅ Velocity: If a car moves 60 km north in 1 hour, the velocity is represented by an arrow: bash ↑ 60 km/h

💡 Key Points:

✔️ Speed = Magnitude only (no direction) ✔️ Velocity = Magnitude + Direction ✔️ Velocity can be zero if the displacement is zero (even if distance is non-zero) ✔️ If an object returns to its starting point, average velocity = zero, but average speed ≠ zero

🚀 Velocity

✅ Definition of Velocity

Velocity is the rate of change of displacement of an object with respect to time in a particular direction. It is a vector quantity because it includes both: Magnitude (how fast the object is moving) Direction (the path the object follows) $$\text{Velocity} = \frac{\text{Displacement}}{\text{Time Taken}}$$

🌍 SI Unit of Velocity:

Meters per second (m/s) Other common units: Kilometer per hour (km/h) Miles per hour (mph)

🏆 Types of Velocity

🔹 1. Uniform Velocity

When an object covers equal displacement in equal intervals of time in a straight line. 📌 Example: A car moving at 60 km/h toward the north without changing speed or direction. 📏 Formula: $$\text{Velocity} = \frac{\text{Displacement}}{\text{Time}}$$ ✅ Graph: A straight line on a velocity-time graph indicates uniform velocity. markdown | ------ v | / e | / l |_______/_________ o t

🔹 2. Non-Uniform (Variable) Velocity

When an object covers unequal displacement in equal intervals of time or changes direction while moving. 📌 Example: A car moving through traffic, changing speed and direction frequently. 📏 Formula: Average Velocity: $$\bar{v} = \frac{\Delta x}{\Delta t}$$ ✅ Graph: A curved line on a velocity-time graph indicates non-uniform velocity. lua | ------ v | / e | / l |______/ o t

🔹 3. Instantaneous Velocity

The velocity of an object at a specific moment of time. 📌 Example: The speedometer of a car shows the instantaneous velocity at a given time. 📏 Formula: $$v_{\text{inst}} = \frac{dx}{dt}$$ ✅ Graph: The slope of a displacement-time graph gives the instantaneous velocity. perl | / x | / | / |__/ t

🔹 4. Average Velocity

The total displacement divided by the total time taken. 📌 Example: A person walks 10 m east and then 10 m west in 10 seconds. Total displacement = 0 m → Average velocity = 0 m/s 📏 Formula: $$\bar{v} = \frac{\text{Total Displacement}}{\text{Total Time}}$$ ✅ Graph: If displacement is zero, the average velocity is zero. markdown |----|----|----| d | |_________________ t

🔹 5. Relative Velocity

The velocity of an object as observed from another moving object. 📌 Example: Two cars moving toward each other at 60 km/h each will have a relative velocity of 120 km/h. 📏 Formula: If two objects A and B are moving at velocities $v_A$ and $v_B$, then: Same direction: $$v_{\text{rel}} = v_A - v_B$$ Opposite direction: $$v_{\text{rel}} = v_A + v_B$$ ✅ Graph: If both objects move toward each other, their velocities are added.

🔢 Numerical Examples

Example 1:

A car moves 120 km toward the east in 2 hours. Find the velocity of the car. 📏 Solution: Given: Displacement = 120 km (east) Time = 2 hours Using the formula: $$v = \frac{\text{Displacement}}{\text{Time}} = \frac{120 \, \text{km}}{2 \, \text{h}} = 60 \, \text{km/h (east)}$$ ✅ Answer: The velocity of the car is 60 km/h east.

Example 2:

A train travels 500 m north in 20 seconds. Find the velocity of the train. 📏 Solution: Given: Displacement = 500 m (north) Time = 20 s Using the formula: $$v = \frac{500 \, \text{m}}{20 \, \text{s}} = 25 \, \text{m/s (north)}$$ ✅ Answer: The velocity of the train is 25 m/s north.

Example 3:

A car is moving at 80 km/h toward the east. Another car is moving at 60 km/h toward the west. Find their relative velocity. 📏 Solution: Given: $v_1 = 80 \, \text{km/h (east)}$ $v_2 = 60 \, \text{km/h (west)}$ Since they are moving toward each other: $$v_{\text{rel}} = 80 + 60 = 140 \, \text{km/h}$$ ✅ Answer: The relative velocity is 140 km/h.

📈 Graphical Representation

✅ Uniform Velocity Graph:

Straight line indicates constant velocity. lua | v | ----- e | / l |_______/ o t

✅ Non-Uniform Velocity Graph:

Curved line indicates changing velocity. perl | v | / e | / l | / o |__/ t

✅ Instantaneous Velocity Graph:

Slope at a specific point gives instantaneous velocity. perl | / x | / | / |___/ t

💡 Key Takeaways:

✔️ Velocity measures both how fast and in what direction an object moves. ✔️ Uniform velocity = constant speed and direction. ✔️ Non-uniform velocity = changing speed or direction. ✔️ Instantaneous velocity = velocity at a particular instant. ✔️ Relative velocity = velocity of an object compared to another moving object.

🚀 Acceleration

✅ Definition of Acceleration

Acceleration is the rate of change of velocity of an object with respect to time. It measures how quickly an object speeds up, slows down, or changes direction. It is a vector quantity because it has both: Magnitude (how much the velocity changes) Direction (the direction of change) $$a = \frac{\Delta v}{\Delta t}$$ where: $a$ = acceleration $\Delta v$ = change in velocity = $v_f - v_i$ $\Delta t$ = time taken for the change in velocity

🌍 SI Unit of Acceleration:

Meters per second squared (m/s²)

🏆 Types of Acceleration

🔹 1. Uniform Acceleration

When an object changes its velocity by an equal amount in equal intervals of time. 📌 Example: A freely falling object under gravity experiences uniform acceleration of 9.81 m/s² downward. 📏 Formula: $$a = \frac{v_f - v_i}{t}$$ ✅ Graph: A straight line on a velocity-time graph shows uniform acceleration. markdown v | | / | / |______/ t

🔹 2. Non-Uniform (Variable) Acceleration

When an object changes its velocity by an unequal amount in equal intervals of time. 📌 Example: A car accelerating at different rates in traffic. 📏 Formula: Instantaneous Acceleration: $$a_{\text{inst}} = \frac{dv}{dt}$$ ✅ Graph: A curved line on a velocity-time graph shows non-uniform acceleration. markdown v | | / | / |___/ t

🔹 3. Positive Acceleration

When an object's velocity increases over time. 📌 Example: A car speeding up from 20 km/h to 60 km/h. ✅ Graph: A positive slope indicates positive acceleration. markdown v | | / | / |______/ t

🔹 4. Negative Acceleration (Deceleration or Retardation)

When an object's velocity decreases over time. 📌 Example: A car slowing down when approaching a red light. ✅ Graph: A negative slope indicates negative acceleration. markdown v | | / | / |____/ t

🔹 5. Centripetal Acceleration

When an object moves in a circular path at a constant speed but continuously changes direction toward the center of the circle. 📌 Example: A satellite orbiting the Earth. 📏 Formula: $$a_c = \frac{v^2}{r}$$ where: $a_c$ = centripetal acceleration $v$ = velocity $r$ = radius of the circular path

📏 Formula for Acceleration

✅ Basic Formula:

$$a = \frac{v_f - v_i}{t}$$ where: $a$ = acceleration $v_f$ = final velocity $v_i$ = initial velocity $t$ = time

🔢 Numerical Examples

Example 1:

A car starts from rest and reaches a speed of 25 m/s in 5 seconds. Find the acceleration. 📏 Solution: Given: $v_i = 0$ (car starts from rest) $v_f = 25 \, m/s$ $t = 5 \, s$ Using the formula: $$a = \frac{v_f - v_i}{t} = \frac{25 - 0}{5} = 5 \, m/s^2$$ ✅ Answer: The acceleration is 5 m/s².

Example 2:

A train is moving at 30 m/s. It comes to rest in 10 seconds. Find the acceleration. 📏 Solution: Given: $v_i = 30 \, m/s$ $v_f = 0 \, m/s$ $t = 10 \, s$ Using the formula: $$a = \frac{0 - 30}{10} = -3 \, m/s^2$$ ✅ Answer: The negative sign indicates deceleration of 3 m/s².

Example 3:

An object starts from rest and accelerates uniformly at 2 m/s² for 4 seconds. Find the final velocity and distance traveled. 📏 Solution: Given: $v_i = 0$ $a = 2 \, m/s^2$ $t = 4 \, s$ (a) Final velocity: Using the first equation of motion: $$v_f = v_i + a t = 0 + (2)(4) = 8 \, m/s$$ (b) Distance traveled: Using the second equation of motion: $$s = v_i t + \frac{1}{2} a t^2$$ $$s = 0 + \frac{1}{2} (2) (4^2) = \frac{1}{2} (2)(16) = 16 \, m$$ ✅ Answer: Final velocity = 8 m/s Distance traveled = 16 m

Example 4:

An object moving at 20 m/s comes to rest over a distance of 50 m. Find the acceleration. 📏 Solution: Given: $v_i = 20 \, m/s$ $v_f = 0 \, m/s$ $s = 50 \, m$ Using the third equation of motion: $$v_f^2 = v_i^2 + 2 a s$$ $$0 = (20)^2 + 2 (a)(50)$$ $$0 = 400 + 100a$$ $$100a = -400$$ $$a = -4 \, m/s^2$$ ✅ Answer: The acceleration is −4 m/s² (negative sign indicates deceleration).

💡 Key Takeaways:

✔️ Acceleration measures how quickly velocity changes. ✔️ Positive acceleration → Speed increases. ✔️ Negative acceleration → Speed decreases (deceleration). ✔️ Constant slope = Uniform acceleration. ✔️ Changing slope = Non-uniform acceleration.

🚀 Retardation (Deceleration)

✅ Definition of Retardation

Retardation (or deceleration) is the rate of decrease of velocity of an object with respect to time. It is essentially negative acceleration because the velocity decreases over time. It is a vector quantity since it has both: Magnitude (how much the velocity decreases) Direction (opposite to the motion of the object) $$a = \frac{\Delta v}{t}$$ where: $a$ = retardation (negative acceleration) $\Delta v$ = change in velocity $t$ = time taken for the change in velocity

🌍 SI Unit of Retardation:

Meters per second squared (m/s²)

🏆 Difference Between Acceleration and Retardation

Property Acceleration Retardation Definition Rate of increase of velocity Rate of decrease of velocity Sign Positive (+) Negative (−) Effect on Object Speeds up the object Slows down the object Example A car increasing speed from 20 km/h to 60 km/h A car slowing down from 60 km/h to 20 km/h Graph Positive slope on velocity-time graph Negative slope on velocity-time graph

🧪 Formula for Retardation

Since retardation is negative acceleration, the basic formula becomes: $$a = \frac{v_f - v_i}{t}$$ where: $a$ = retardation (negative value) $v_f$ = final velocity $v_i$ = initial velocity $t$ = time taken If the velocity decreases, the value of $v_f - v_i$ becomes negative, resulting in negative acceleration (retardation).

🔢 Numerical Examples

Example 1:

A car moving at 30 m/s slows down uniformly to 10 m/s in 5 seconds. Find the retardation. 📏 Solution: Given: $v_i = 30 \, m/s$ $v_f = 10 \, m/s$ $t = 5 \, s$ Using the formula: $$a = \frac{v_f - v_i}{t} = \frac{10 - 30}{5} = \frac{-20}{5} = -4 \, m/s^2$$ ✅ Answer: Retardation = −4 m/s² (negative sign indicates a decrease in speed).

Example 2:

A train traveling at 40 m/s stops in 20 seconds. Find the retardation. 📏 Solution: Given: $v_i = 40 \, m/s$ $v_f = 0 \, m/s$ (since the train stops) $t = 20 \, s$ Using the formula: $$a = \frac{0 - 40}{20} = \frac{-40}{20} = -2 \, m/s^2$$ ✅ Answer: Retardation = −2 m/s²

Example 3:

A car is traveling at 25 m/s and comes to rest after covering a distance of 100 m. Find the retardation. 📏 Solution: Given: $v_i = 25 \, m/s$ $v_f = 0 \, m/s$ $s = 100 \, m$ Using the third equation of motion: $$v_f^2 = v_i^2 + 2 a s$$ $$0 = (25)^2 + 2 (a) (100)$$ $$0 = 625 + 200a$$ $$200a = -625$$ $$a = -3.125 \, m/s^2$$ ✅ Answer: Retardation = −3.125 m/s²

💡 Key Takeaways:

✔️ Retardation is negative acceleration. ✔️ It occurs when an object slows down over time. ✔️ It always acts in the opposite direction to the motion of the object. ✔️ Negative slope in velocity-time graph = Retardation.

🚀 Average Velocity and Relative Velocity

✅ 1. Average Velocity

➡️ Definition of Average Velocity

Average velocity is defined as the total displacement divided by the total time taken. $$\bar{v} = \frac{\Delta x}{\Delta t}$$ where: $\bar{v}$ = average velocity $\Delta x$ = total displacement $\Delta t$ = total time taken

🌍 SI Unit of Average Velocity:

Meters per second (m/s)

🏆 Example of Average Velocity:

A car travels 120 km north in 2 hours and then returns 60 km south in 1 hour. Find the average velocity. 📏 Solution: Total displacement = $120 \, \text{km} - 60 \, \text{km} = 60 \, \text{km}$ (north) Total time = $2 + 1 = 3 \, \text{hours}$ $$\bar{v} = \frac{60 \, \text{km}}{3 \, \text{h}} = 20 \, \text{km/h}$$ ✅ Answer: The average velocity = 20 km/h (north)

🏆 Average Speed vs Average Velocity:

Property Average Speed Average Velocity Definition Total distance traveled per unit time Total displacement per unit time Formula $\frac{\text{Total distance}}{\text{Total time}}$ $\frac{\text{Total displacement}}{\text{Total time}}$ Value Always positive Can be positive, negative, or zero Example A car moving in a circle → High average speed, zero average velocity A car moving in a straight line → Both are equal

✅ 2. Relative Velocity

➡️ Definition of Relative Velocity

Relative velocity is the velocity of one object with respect to another object. It represents how fast one object is moving compared to another. $$\vec{v}_{AB} = \vec{v}_A - \vec{v}_B$$ where: $\vec{v}_{AB}$ = velocity of object A relative to object B $\vec{v}_A$ = velocity of object A $\vec{v}_B$ = velocity of object B

🌍 SI Unit of Relative Velocity:

Meters per second (m/s)

🏆 Types of Relative Velocity:

🔹 (a) Objects Moving in the Same Direction

If two objects are moving in the same direction, the relative velocity is the difference of their velocities: $$v_{AB} = v_A - v_B$$ 📌 Example: Object A = 40 m/s Object B = 30 m/s $$v_{AB} = 40 - 30 = 10 \, m/s$$ ✅ Answer: The relative velocity = 10 m/s in the direction of object A.

🔹 (b) Objects Moving in Opposite Directions

If two objects are moving in opposite directions, the relative velocity is the sum of their velocities: $$v_{AB} = v_A + v_B$$ 📌 Example: Object A = 40 m/s (right) Object B = 30 m/s (left) $$v_{AB} = 40 + 30 = 70 \, m/s$$ ✅ Answer: The relative velocity = 70 m/s

🔹 (c) Object at Rest

If one object is at rest, the relative velocity is simply the velocity of the moving object. 📌 Example: Object A = 50 m/s Object B = 0 m/s (rest) $$v_{AB} = 50 - 0 = 50 \, m/s$$ ✅ Answer: The relative velocity = 50 m/s

🏆 Example of Relative Velocity:

Two cars A and B are moving in opposite directions. Car A is moving at 60 km/h east and car B is moving at 40 km/h west. Find the relative velocity of A with respect to B. 📏 Solution: Since they are moving in opposite directions: $$v_{AB} = v_A + v_B = 60 + 40 = 100 \, \text{km/h}$$ ✅ Answer: The relative velocity = 100 km/h

🔢 Numerical Problems

Example 1:

A train moving at 100 m/s overtakes another train moving at 60 m/s in the same direction. What is the relative velocity of the faster train with respect to the slower one? 📏 Solution: $v_A = 100 \, m/s$ $v_B = 60 \, m/s$ Since they are moving in the same direction: $$v_{AB} = 100 - 60 = 40 \, m/s$$ ✅ Answer: Relative velocity = 40 m/s

Example 2:

Two cars A and B are moving towards each other. Car A is moving at 80 m/s and car B is moving at 60 m/s. Find the relative velocity. 📏 Solution: Since they are moving in opposite directions: $$v_{AB} = 80 + 60 = 140 \, m/s$$ ✅ Answer: Relative velocity = 140 m/s

🔬 Difference Between Average Velocity and Relative Velocity

Property Average Velocity Relative Velocity Definition  Total displacement per unit time Velocity of one object relative to another Formula $\bar{v} = \frac{\Delta x}{\Delta t}$ $v_{AB} = v_A - v_B$ Direction Same as the direction of displacement Depends on the motion of both objects Value A single value over a time interval Can be positive, negative, or zero Example Car covering 100 km in 2 hours → Average velocity = 50 km/h Two trains moving in opposite directions at 40 km/h and 30 km/h → Relative velocity = 70 km/h

💡 Key Takeaways:

✔️ Average velocity → Total displacement over time. ✔️ Relative velocity → How fast one object appears to move relative to another. ✔️ Relative velocity depends on the direction of motion. ✔️ Average velocity depends on displacement and time.