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HSSLIVE Plus One Physics Chapter 3: Motion in a Straight Line Notes

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Motion in a Straight Line introduces students to kinematics, the study of how objects move without considering the forces causing the motion. This chapter covers position, displacement, velocity, and acceleration—using graphs and equations to describe one-dimensional motion. Students master uniform and non-uniform motion concepts, applying calculus-based relationships to solve practical problems involving straight-line trajectories.

Chapter 3: Motion in a Straight Line

Plus One Physics – SCERT Kerala Board

1. Introduction to Motion

  • Definition: Motion is the change in position of an object with respect to time.
  • Types of Motion: Translational, rotational, vibrational, random, etc.
  • Frame of Reference: A system of coordinates with respect to which motion is measured.
  • One-dimensional motion: Motion along a straight line (along one coordinate axis).

2. Position, Path Length and Displacement

  • Position: Location of an object described by coordinates relative to a reference point.
  • Path Length: Total distance traveled by an object along its path.
  • Displacement: Change in position (vector quantity).
    • Displacement = Final position – Initial position
    • Formula: $\vec{s} = \vec{r}_f – \vec{r}_i$
  • Difference between path length and displacement:
    • Path length is always positive and scalar
    • Displacement can be positive, negative, or zero and is a vector
    • Path length ≥ Magnitude of displacement

3. Average Velocity and Average Speed

  • Average Speed: Total path length divided by total time taken.
    • Formula: $v_{avg} = \frac{\text{Total path length}}{\text{Total time taken}} = \frac{s}{t}$
    • SI unit: meter per second (m/s)
  • Average Velocity: Displacement divided by time interval.
    • Formula: $\vec{v}_{avg} = \frac{\Delta \vec{s}}{\Delta t} = \frac{\vec{s}_f – \vec{s}_i}{t_f – t_i}$
    • SI unit: meter per second (m/s)
  • Difference: Average speed considers path length while average velocity considers displacement.

4. Instantaneous Velocity and Speed

  • Instantaneous Velocity: Velocity at a specific instant of time.
    • Formula: $\vec{v} = \lim_{\Delta t \to 0} \frac{\Delta \vec{s}}{\Delta t} = \frac{d\vec{s}}{dt}$
    • Direction: Same as the direction of instantaneous displacement
  • Instantaneous Speed: Magnitude of instantaneous velocity.
    • Formula: $v = |\vec{v}| = \left|\frac{d\vec{s}}{dt}\right|$

5. Acceleration

  • Definition: Rate of change of velocity with respect to time.
  • Average Acceleration: Change in velocity divided by time interval.
    • Formula: $\vec{a}_{avg} = \frac{\Delta \vec{v}}{\Delta t} = \frac{\vec{v}_f – \vec{v}_i}{t_f – t_i}$
  • Instantaneous Acceleration: Acceleration at a specific instant of time.
    • Formula: $\vec{a} = \lim_{\Delta t \to 0} \frac{\Delta \vec{v}}{\Delta t} = \frac{d\vec{v}}{dt}$
  • SI unit: meter per second squared (m/s²)
  • Sign of acceleration:
    • Positive: When velocity and acceleration have the same direction
    • Negative: When velocity and acceleration have opposite directions

6. Kinematic Equations for Uniformly Accelerated Motion

For motion with constant acceleration:

Equation Formula Variables
First equation $v = u + at$ $v$ = final velocity, $u$ = initial velocity, $a$ = acceleration, $t$ = time
Second equation $s = ut + \frac{1}{2}at^2$ $s$ = displacement, $u$ = initial velocity, $a$ = acceleration, $t$ = time
Third equation $v^2 = u^2 + 2as$ $v$ = final velocity, $u$ = initial velocity, $a$ = acceleration, $s$ = displacement
Fourth equation (derived) $s = \frac{1}{2}(u+v)t$ $s$ = displacement, $u$ = initial velocity, $v$ = final velocity, $t$ = time

7. Motion Under Gravity (Free Fall)

  • Definition: Motion of an object under the influence of gravity alone, neglecting air resistance.
  • Acceleration due to gravity (g):
    • Value: approximately 9.8 m/s² near Earth’s surface
    • Direction: Always towards the center of the Earth
  • Equations: Same as uniformly accelerated motion with $a = g$
    • For upward motion: $a = -g$ (negative)
    • For downward motion: $a = g$ (positive)

8. Relative Velocity

  • Definition: Velocity of one object as observed from another object.
  • Formula: $\vec{v}_{AB} = \vec{v}_A – \vec{v}_B$
    • $\vec{v}_{AB}$ = velocity of A relative to B
    • $\vec{v}_A$ = velocity of A relative to ground
    • $\vec{v}_B$ = velocity of B relative to ground

9. Graphical Representation of Motion

9.1 Position-Time Graph

  • Uniform motion: Straight line with slope equal to velocity
  • Non-uniform motion: Curved line
  • Slope at any point: Instantaneous velocity at that point
  • Zero slope: Object at rest

9.2 Velocity-Time Graph

  • Uniform motion: Horizontal line
  • Uniformly accelerated motion: Straight line with slope equal to acceleration
  • Area under the curve: Displacement during the time interval
  • Slope at any point: Instantaneous acceleration at that point

9.3 Acceleration-Time Graph

  • Constant acceleration: Horizontal line
  • Area under the curve: Change in velocity during the time interval

10. Solving Problems on Motion in a Straight Line

10.1 General Approach:

  1. Define coordinate system and positive direction
  2. List known and unknown quantities
  3. Select appropriate kinematic equation(s)
  4. Solve for the unknown quantity
  5. Check the answer for consistency (units and reasonableness)

10.2 Common Types of Problems:

  • Finding displacement given initial velocity, acceleration, and time
  • Finding final velocity given initial velocity, acceleration, and displacement
  • Finding time taken to reach a certain position or velocity
  • Finding acceleration required to achieve certain conditions

11. Important Points to Remember

  • In one-dimensional motion, vectors can be treated as scalars with signs (+ or -) to indicate direction
  • The sign of displacement and velocity depends on the chosen coordinate system
  • For a body thrown upward:
    • Velocity decreases until it becomes zero at maximum height
    • Velocity changes direction at maximum height
    • Magnitude of velocity when it returns to starting point equals initial velocity
  • Average velocity can be zero even when average speed is not zero
  • Instantaneous speed is the magnitude of instantaneous velocity
  • If acceleration is in the direction of velocity, speed increases
  • If acceleration is opposite to velocity, speed decreases

12. Summary

  • Motion in a straight line is described by position, displacement, velocity, and acceleration
  • Average quantities refer to the entire time interval while instantaneous quantities refer to a specific moment
  • Kinematic equations relate position, velocity, acceleration, and time for uniformly accelerated motion
  • Graphical representations help visualize and analyze motion
  • Free fall is a special case of uniformly accelerated motion with acceleration equal to g
  • Relative velocity is important when analyzing motion from different frames of reference.

 

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