motion in one dimension

scalar and vector quantities

physical quantities are measurable and fall into two categories:

• scalar quantities: expressed only by magnitude.- examples: mass, length, time, distance, speed, temperature, work, energy, power, pressure, charge, electric power, resistance, frequency and angle.

  • they can be added, subtracted, multiplied, and divided using simple arithmetic methods.

  • represented by english letters (e.g., m for mass, t for time, v for speed).

• vector quantities: require both magnitude and direction for complete meaning.- examples: displacement, velocity, acceleration, momentum, force, moment of force (torque), impulse, weight, temperature gradient, electric field, magnetic field, and dipole moment.

  • magnitude is always positive; a negative sign indicates the opposite direction.

  • follow different algebra rules for addition, subtraction, and multiplication.

  • written with an arrow above the letter or in bold (e.g., \vec{v} or v for velocity, \vec{a} or a for acceleration, \vec{F} or f for force).

rest and motion

• every object in the universe is in motion. even objects that appear to be at rest are moving with the earth as it orbits the sun, so we often say that a stone lying on the ground is at rest because the stone does not change its position with respect to us.

• rest: a body is at rest if it does not change its position with respect to its immediate surroundings.

• motion: a body is in motion if it changes its position with respect to its immediate surroundings.

• point particle: when the distance traveled by a moving body is much larger than its size, the body can be considered a point particle.

• one-dimensional motion: motion along a straight line.- also called rectilinear motion.

  • example: motion of a train on a straight track, a stone falling vertically.

  • there is no sideways movement.

• two-dimensional motion: motion on a plane along a curved path.

• three-dimensional motion: motion in space.

representation of one-dimensional motion

• represented by a straight line parallel to the x-axis.

• the position of a particle at any instant t is expressed by its x coordinate at that instant.

• example: a pebble falling vertically downwards, with its position x measured from a starting point at different times t.

distance and displacement

• consider a body moving from point a to point b along a path.

• distance: the total length of the path from a to b.- depends on the path followed.

  • scalar quantity represented by s.

  • si unit: metre (m); cgs unit: centimetre (cm).

• displacement: the shortest distance from the initial position (a) to the final position (b), with direction from a to b.- vector quantity represented by \vec{s}.

  • si unit: metre (m); cgs unit: centimetre (cm).

representation of displacement

• represented by a straight line with an arrow, using a convenient scale.

• the arrow indicates direction, and the length represents magnitude.

• example: vector \vec{pq} represents 40 m displacement in the east direction, using a scale of 1 cm = 10 m.

distinction between distance and displacement

1. magnitude:

   • displacement magnitude is either equal to or less than the distance.

   • equal if motion is in a fixed direction; less if motion is along a curve or zig-zag path.

   • displacement magnitude can never be greater than the distance.

2. sign:

   • distance is always positive.

   • displacement can be positive or negative depending on direction.

3. zero values:

   • displacement can be zero even if the distance is not zero (e.g., a body returns to its starting point).

4. definitions:
   

   distance	displacement
   length of the path traversed by the object in a certain time	distance travelled by the object in a specified direction in a certain time (shortest distance between positions)
   scalar quantity (magnitude only)	vector quantity (magnitude and direction)
   depends on the path followed	does not depend on the path followed
   always positive	can be positive or negative
   can be more than or equal to the magnitude of displacement	its magnitude can be less than or equal to the distance, but never greater
   may not be zero even if displacement is zero, but cannot be zero if displacement is not zero	is zero is distance is zero, but can be zero even if the distance is not zero

   • speed: indicates how fast a body is moving.

   • velocity: indicates the speed and direction of motion.

   speed

   • the rate of change of distance with time.

   • numerically, the distance traveled in 1 s.

   • scalar quantity represented by u or v.

   • speed (v) = \frac{distance (s)}{time (t)} \Rightarrow {v = \frac{s}{t}}

   • unit: metre per second (m/s or ms^{-1}) in si, centimetre per second (cm/s or cms^{-1}) in cgs.

     • uniform speed: equal distances covered in equal intervals of time.

       • example: motion of a ball on a frictionless plane surface.

       • distance s = vt

     • non-uniform (variable) speed: unequal distances covered in equal intervals of time.

       • examples: motion of a ball on a rough surface, motion of a car in a crowded street.

     • instantaneous speed: speed at any instant, measured by finding the ratio of distance traveled in a very short time interval to that time interval.

       instantaneous speed = \frac{distance travelled in a short time interval}{time interval}

       • the speedometer of a vehicle measures instantaneous speed.

     • average speed:

       average speed = \frac{total distance travelled}{total time taken}}

   velocity

   • the distance traveled per second in a specified direction.

   • the rate of change of displacement with time.

   • numerically equal to the displacement in 1 s.

   • vector quantity represented by \vec{u} or \vec{v}.

   • unit: metre per second (m/s or ms^{-1}) in si, centimetre per second (cm/s or cms^{-1}) in cgs.

     • uniform velocity: equal distances in a particular direction in equal intervals of time.

       • example: rain drops falling with terminal velocity.

       • displacement \vec{s} = \vec{v}t

     • non-uniform (variable) velocity: velocity changes in magnitude, direction, or both.

       • examples: motion of a freely falling body, motion of a body in a circular path.

     • instantaneous velocity: velocity at any instant.

       instantaneous velocity = \frac{distance travelled in a short time interval}{time interval}

     • average velocity:

       average velocity = \frac{displacement}{total time taken}}

   distinction between speed and velocity

   speed	velocity
   scalar quantity	vector quantity
   indicates how fast a body is moving	indicates how fast the body is moving and the direction
   always positive	can be positive or negative depending on the direction of motion
   average speed can be non-zero	average velocity can be zero, even if the average speed is not

   acceleration and retardation

   • acceleration: rate of change of velocity with time.

     • numerically equal to the change in velocity in 1 s.

     • acceleration = \frac{change in velocity}{time interval}}

     • a = \frac{v-u}{t}, where:

       • a = acceleration

       • v = final velocity

       • u = initial velocity

       • t = time interval

     • unit: metre per second square (m/s² or ms^{-2}) in si, centimetre per second square (cm/s² or cms^{-2}) in cgs.

   • retardation (deceleration): negative acceleration (decrease in velocity with time).

   • acceleration is a vector quantity with direction of change in velocity.

   • the positive or negative sign of acceleration indicates whether the velocity is increasing or decreasing.

   • uniform acceleration: equal changes in velocity in equal intervals of time.

     • example: free fall of a body under gravity.

   • variable acceleration: change in velocity is not the same in equal intervals of time.

     • example: motion of a vehicle on a crowded road.

   • acceleration due to gravity (g): acceleration produced in a body due to earth's gravitational attraction, approximately 9.8 m/s².

     • value varies from place to place (maximum at poles, minimum at equator).

   graphical representation of linear motion

   graphical methods for studying linear motion:

   • displacement-time graph.

   • velocity-time graph.

   • acceleration-time graph.

   • in one-dimensional motion, the displacement-time graph is the same as the distance-time graph, and the velocity-time graph is the same as the speed-time graph.

   displacement-time graph

   • time is taken on the x-axis, and displacement on the y-axis.

   • the slope of the displacement-time graph gives the velocity.

     • positive slope indicates motion away from the starting point.

     • negative slope indicates motion towards the starting point.

     • if the graph is a straight line parallel to the time axis, the object is stationary.

     • if the graph is a straight line inclined to the time axis, the motion is with uniform velocity.

     • if the graph is a curve, the motion is with non-uniform velocity.

   velocity-time graph

   • time is taken on the x-axis, and the velocity is taken on the y-axis.

   • positive velocity indicates motion in a certain direction, and negative velocity indicates motion in the opposite direction.

     • the area enclosed between the velocity-time sketch and x-axis gives the displacement.

     • the area above the time axis represents positive displacement, and the area below represents negative displacement.

     • the slope of the velocity-time sketch gives the acceleration.

     • if the graph is a straight line parallel to the time axis, the motion is with uniform velocity.

     • if the graph is a straight line inclined to the time axis, the motion is with uniform acceleration.

     • if the graph is a curve, the motion is with non-uniform acceleration.

   acceleration-time graph

   • time is taken on the x-axis, and acceleration is taken on the y-axis.

   • the area enclosed between the acceleration-time sketch and the time axis gives the change in speed.

     • if the body is stationary or moving with a uniform velocity, the acceleration is zero, and the graph is a straight line coinciding with the time axis.

     • if the velocity of the body increases uniformly with time, the acceleration is constant, and the graph is a straight line parallel to the time axis on the positive acceleration axis.

     • if the velocity of the body decreases at a constant rate, the retardation is constant, and the graph is a straight line parallel to the time axis on the negative acceleration axis.

   motion under gravity

   • a body falling freely under gravity moves with a uniform acceleration of approximately 9.8 m/s².

   • for a body moving vertically upwards, there is a uniform retardation.

   equations of uniformly accelerated motion

   • for a body moving with uniform acceleration, the relationship between initial velocity (u), final velocity (v), acceleration (a), time (t), and distance traveled (s) is:

     1. v = u + at

     2. s = ut + \frac{1}{2} a t^2

     3. v^2 = u^2 + 2 a s

   • if a body starts from rest (u = 0):

     1. v = at

     2. s = \frac{1}{2} a t^2

     3. v^2 = 2 a s

   • if a body is under uniform retardation, a is negative:

     1. v = u - at

     2. s = ut - \frac{1}{2} a t^2