Kinematics

Meaning and Definitions

• Position: The location of an object relative to a reference point (the "origin").
• Distance (Scalar): The total path length traveled by an object. It doesn't care about direction.
• Displacement (Vector): The shortest straight-line distance between the starting point and the ending point, including the direction.

Explanation

Imagine you walk 5 meters East, then 5 meters West. Your distance is 10 meters (total path), but your displacement is 0 meters because you returned to the start.

Key Points

• Distance is always positive and ≥ |Displacement|.
• Displacement can be positive, negative, or zero.

Examples

• The Track Runner: A runner completes one full lap of a 400m circular track. Distance = 400m; Displacement = 0m.
• The Commuter: A car drives 10km North and then 3km South. Distance = 13km; Displacement = 7km North.

Real-Life Use: GPS Navigation

Google Maps uses distance to estimate fuel consumption, but uses displacement relative to destination coordinates to confirm you have actually arrived at the correct spot.

Meaning and Definitions

• Speed (Scalar): The rate at which an object covers distance.
  Formula: Speed = Distance / Time
• Velocity (Vector): The rate of change of displacement.
  Formula: Velocity = Displacement / Time

Comparison Table: Speed vs. Velocity

Examples

• Constant Speed: A car moves at a steady 60 km/h on a straight highway. Speed and velocity are identical.
• Changing Velocity: A car driving in a circle at 40 km/h has constant speed, but its velocity is constantly changing due to direction change.

Real-Life Use: Speed Cameras vs. Radar Guns

Speed cameras measure speed, whereas police radar guns often measure velocity to ensure correct direction on one-way streets.

Meaning and Definitions

• Uniform Motion: Covering equal distances in equal time intervals (constant velocity).
• Non-Uniform Motion: Covering unequal distances in equal time intervals (changing velocity).
• Acceleration (Vector): The rate of change of velocity.
  Formula: a = (v - u) / t

Key Points

• Positive acceleration = speeding up.
• Negative acceleration (deceleration/retardation) = slowing down.

Real-Life Use: Elevator Design

Engineers calculate acceleration to ensure elevator movement is smooth and doesn't cause passengers to feel uncomfortable "jerkiness."

The Formulas

1. First Equation: v = u + at
2. Second Equation: s = ut + ½at²
3. Third Equation: v² = u² + 2as

Where: u = initial velocity, v = final velocity, a = acceleration, t = time, s = displacement.

Numerical Illustration

Problem: A car traveling at 30 m/s applies brakes and comes to a stop over 60 m. Find acceleration and time.

1. Given: u = 30 m/s, v = 0 m/s, s = 60 m
2. Find 'a' using v² = u² + 2as:
   0² = (30)² + 2(a)(60)
   0 = 900 + 120a
   a = -900 / 120 = -7.5 m/s² (Deceleration)
3. Find 't' using v = u + at:
   0 = 30 + (-7.5)t
   7.5t = 30
   t = 4 seconds
        

Scalar vs. Vector

• Scalars: Magnitude only (Mass, Temp, Speed).
• Vectors: Magnitude + Direction (Force, Velocity, Acceleration).

Projectile Motion

An object thrown into the air following a parabolic path. It has a constant horizontal velocity and constant vertical acceleration (gravity).

Key Projectile Formulas

• Time of Flight (T) = (2u sin θ) / g
• Max Height (H) = (u² sin² θ) / 2g
• Horizontal Range (R) = (u² sin 2θ) / g

Uniform Circular Motion

Moving in a circle at constant speed implies constant Centripetal Acceleration because the direction is always changing.

The velocity of an object as observed from another moving object.

Formula: v_AB = v_A - v_B

Example: Two cars moving in the same direction at 100 km/h and 80 km/h have a relative velocity of 20 km/h.

An automated drone performing a delivery utilizes all kinematic principles. During takeoff, it undergoes acceleration. While flying at a steady altitude, it moves in uniform motion. To combat crosswinds, the software calculates relative velocity. Finally, its landing approach is a controlled projectile-like descent using negative acceleration to ensure a soft stop at the origin.

1. The slope of a Position-Time graph represents:
   (b) Velocity
2. Which of the following is a vector quantity?
   (c) Displacement
3. If an object has constant velocity, its acceleration is:
   (c) Zero
4. A body moving in a circle with constant speed has:
   (c) Changing velocity
5. The area under a Velocity-Time graph gives:
   (b) Displacement
6. A ball is thrown upwards. At its highest point, its velocity is:
   (c) Zero
7. If distance is 10m and displacement is 10m, the motion is:
   (b) In a straight line
8. The unit of acceleration is:
   (c) m/s²
9. Projectile motion is a combination of:
   (b) Horizontal and Vertical motion
10. Relative velocity of two objects moving in opposite directions is:
    (b) V₁ + V₂

1. Distinguish between Distance and Displacement with an example.
   Distance is total path length (scalar), while displacement is the shortest path between start and end (vector). Example: A full lap on a track has distance = circumference, but displacement = 0.
2. Derive the equation v = u + at.
   a = (v - u) / t → at = v - u → v = u + at.
3. Why is circular motion at constant speed considered accelerating?
   Velocity is a vector. In a circle, direction changes continuously. A change in direction means a change in velocity, which is acceleration.
4. A stone is dropped from a tower. What is its initial velocity and acceleration?
   u = 0 m/s; a = 9.8 m/s² (gravity).
5. When is displacement equal to distance?
   When motion is in a single straight line without changing direction.
6. Define Projectile Motion and its factors.
   Motion under gravity alone. Factors: Initial velocity, Angle, and Gravity.
7. How do you find velocity from a Position-Time graph?
   By calculating the slope (gradient) of the line.
8. Relative velocity of two trains moving in the same direction at 60km/h and 40km/h?
   60 - 40 = 20 km/h.
9. Write the three equations of motion.
   v = u + at; s = ut + ½at²; v² = u² + 2as.
10. What angle maximizes projectile range?
    45 degrees.

• Kinematic Equations: v = u + at | s = ut + ½at² | v² = u² + 2as
• Scalar vs Vector: Scalar = Size only | Vector = Value + Via (direction)
• Projectile Max Range: Occurs at 45°.

Mnemonic for Kinematics: Remember "V-U-A-T" (Velocity = u + at).