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Physics – Classical Mechanics

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    Study Guide Physics – Classical Mechanics 1. Kinematics in One Dimension Kinematics is the branch of physics that studies how objects move , without worrying about why they move. In this chapter, we focus on motion along a straight line and explore the relationships between: • Displacement (d) • Velocity (v) • Acceleration (a) • Time (t) Among these, displacement, velocity, and acceleration are vector quantities , which means they include both size and direction. 1. Vectors and Scalars What is a Vector? A vector is a physical quantity that has: • Magnitude (how much), and • Direction (which way) Examples of vectors: • Velocity • Displacement • Force Vectors are often shown using arrows . The length of the arrow represents magnitude, and the arrowhead shows direction.

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    Study Guide What is a Scalar? A scalar has only magnitude and no direction . Examples of scalars: • Time • Temperature • Distance 1 . Displacement and Velocity Imagine a car starting from a signpost on a road. To describe where the car is later, we must know: • How far it moved , and • In which direction it moved 2 . Displacement Displacement is the change in position of an object. It is a vector, so it includes direction. Example: • 10 km east • −5 m (meaning 5 m in the opposite direction) Displacement is not the same as distance traveled. If a car travels: • 1 km east, then • 1 km west back to its starting point The total displacement is zero , even though the car traveled 2 km.

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    Study Guide 3 . Average Velocity Velocity describes how fast an object moves and in which direction . The average velocity is defined as: Units: meters per second (m/s) 4 . Average Acceleration Acceleration tells us how quickly velocity changes over time. It is calculated using: Units: • m/s² (meters per second squared) Acceleration can be: • Positive (speeding up in the positive direction) • Negative (slowing down or speeding up in the opposite direction) 2. Graphs in Kinematics Graphs help us visualize motion and understand relationships between displacement, velocity, and acceleration. 1. Distance – Time Graph A distance – time graph shows how far an object moves as time passes. • A horizontal line → object is at rest

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    Study Guide • A straight sloping line → constant velocity • A steeper slope → higher speed The slope of the graph gives the speed . 2. Velocity – Time Graph A velocity – time graph shows how velocity changes with time. • The slope of this graph gives acceleration • The area under the graph gives displacement For example: • A triangular area represents increasing or decreasing velocity • A rectangular area represents constant velocity

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    Study Guide 3. Instantaneous Velocity and Acceleration 1. Instantaneous Velocity This is the velocity at a specific moment in time — like what a car’s speedometer shows. On a graph: • It is found by drawing a tangent to the distance – time curve • The slope of the tangent gives the instantaneous velocity 2. Instantaneous Acceleration Instantaneous acceleration is found in a similar way: • Draw a tangent to the velocity – time graph • The slope of that tangent gives acceleration at that moment 3 . Connecting the Three Graphs When distance – time, velocity – time, and acceleration – time graphs are placed one above the other:

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    Study Guide • You can find displacement, velocity, and acceleration at the same instant • This makes analyzing motion much easier 5. Negative Acceleration (Deceleration) Negative acceleration does not always mean slowing down . It can happen in two cases: • Case 1: Velocity is positive, but decreasing • Case 2: Velocity is negative, but increasing in magnitude Example: A ball thrown upward: • Slows down while moving upward

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    Study Guide • Stops momentarily at the highest point • Speeds up downward due to gravity 6. Motion with Constant Acceleration When acceleration is constant: • Velocity changes at a steady rate • Average acceleration equals instantaneous acceleration Using: • Initial velocity ( v ₀ ) • Final velocity ( v ᶠ ) • Acceleration ( a ) • Time ( t ) • Displacement ( d ) We can describe motion using four key equations. 7. Kinematic Equations (One Dimension) 1. v ᶠ = v ₀ + at Velocity as a function of time 2. d = ½ (v ₀ + v ᶠ )t Displacement using average velocity 3. d = v ₀ t + ½ at² Displacement as a function of time

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    Study Guide 4. v ᶠ ² = v ₀ ² + 2ad Velocity as a function of displacement Each equation uses four of the five variables , allowing you to solve problems efficiently. 2. Kinematics in Two Dimensions In real life, motion usually happens in more than one direction. To make problems easier to understand, we often describe motion using two dimensions , usually the x - direction (horizontal) and y - direction (vertical) . Examples of two - dimensional motion include: • A ball thrown into the air • A projectile moving across the ground

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    Study Guide • An object moving in a circular path To study this kind of motion, we use vectors and basic vector algebra. 1. Vector Addition and Subtraction: Geometric Method Vectors can be moved anywhere on a page as long as their length and direction stay the same . This allows us to add and subtract them graphically. 1. Adding Two Vectors Suppose: • Vector A represents a velocity of 10 m/s toward the northeast • Vector B represents a velocity of 20 m/s at 30° north of east To add vectors graphically: 1. Place the tail of one vector at the head of the other 2. The resultant vector (C) goes from the tail of the first vector to the head of the second The length of the resultant gives its magnitude, and the angle it makes with the horizontal gives its direction.

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    Study Guide Order of Addition Does Not Matter Vector addition follows the commutative property : • A + B = B + A When vectors are placed tail - to - tail, the resultant is the diagonal of a parallelogram formed by the two vectors. Adding Several Vectors When adding more than two vectors: • Place them head - to - tail , one after another • The resultant is the vector that closes the polygon 2. Subtracting Vectors To subtract vectors: • Place their tails together

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