Introduction

  1. If a body is in uniform motion (velocity = constant or body has acceleration and velocity acting at 0° or 180°, then path followed is straight line and motion is rectilinear motion.

  2. Uniform motion

    \to Velocity is constant and acceleration is zero.

    \to Body covers equal displacement in equal time interval.

    \to Equation for uniform motion s = ut .

    \to time-displacement graph is straight line .

    \to Time velocity graph is straight line parallel to time axis

    \to Time acceleration graph is straight line coincide it to time axis.

  3. Variable motion at constant acceleration

    \to Velocity changes at constant rate and acceleration is constant.

    \to Body may cover equal distances in equal time intervals but never covers equal displacements in equal time interval (a and v at 180°)

  4. Distance and Displacement

    \to The length of actual path followed by a moving body is called distance. It is a scalar quantity.

    \to The shortest distance between initial and final position with direction for a moving body is called displacement . It is a vector quantity.

    \to Displacement covered by a moving body in certain time interval may be zero, negative or positive but distance never be zero or negative.

    \to Distance ≥ Displacement

    \to \dfrac{\text{Distance}}{\text{Displacement}} ≥ 1

    \to For a moving body, the distance travelled always increases with time displacement may increase or decrease with time.

Speed and Velocity

  1. Speed
  2. \to The time rate of change in distance for a moving body is called speed. It is a scalar quantity.

    \to Speed =\dfrac{\text{distance}}{\text{time taken}}

    \to The speed of a body can be zero or positive but never negative.

    \to A body is said to be moving with uniform speed if it covers equal distances in equal intervals of time.

    \to A body is said to be moving with a variable speed if it covers equal distances in unequal intervals of time or unequal distances in equal intervals of time.

    \to Average speed = \dfrac{\text{distance covered in given time}}{\text{time taken}}

    V_{av} = \dfrac{(x-x_0)}{(t-t_0)} = \dfrac{∆x}{∆t}

    Instantaneous speed is limiting value of average speed as time interval tends to zero.

    V_{av} = \displaystyle{\lim_{∆t \to 0}} \dfrac{∆x}{∆t} = \dfrac{dx}{dt}

    It is defined for a point of time .

  3. Velocity

    \to The time rate of change in displacement tor a moving body is called velocity . It is a vector quantity .

    \to Velocity=\dfrac{\text{displacement}}{\text{time taken}}

    \to Average velocity = \dfrac{\text{displacement covered in given time}}{\text{time taken}}

    V_{av} = \dfrac{(s-s_0)}{(t-t_0)} = \dfrac{∆s}{∆t}

    It is defined for period of time.

    Instantaneous speed is limiting value of average speed as time interval tends to zero.

    V= \displaystyle{\lim_{∆t \to 0}} \dfrac{∆s}{∆t} = \dfrac{ds}{dt}

    It is defined for a point of time .

  4. Average velocity and Average speed

    \to Average velocity for a moving body in certain time interval may be zero, +ve or -ve but average speed can never be zero or –ve .

    \to Average speed ≥ Average velocity

    \to \dfrac{\text{Average speed}}{\text{Average velocity}} ≥1

  5. Average velocity

    When a body moving along straight line covers distances s_1 , s_2 , . . . s_n at different velocities v_1 , v_2, ... v_n then

    Average velocity = \dfrac{\text{Total displacement}}{\text{Total time}}

    V_{av}= \dfrac{(s_1 +s_2+ , . . .+ s_n)}{(t_1 + t_2+ .. . . .+ t_n)}

    V_{av}= \dfrac{(s_1 +s_2+ , . . .+ s_n)}{(\dfrac{s_1}{v_1} + \dfrac{s_2}{v_2}+ .. . . .+ \dfrac{s_n}{v_n})}

    \dfrac{(s_1 +s_2+ , . . .+ s_n)}{V_{av}} = \dfrac{s_1}{v_1} + \dfrac{s_2}{v_2}+ .. . . .+ \dfrac{s_n}{v_n}

  6. Equal Distance

    If s_1=s_2 . . . . . =s_n when body covers equal distances then ,

    \dfrac{(ns)}{V_{av}} = \dfrac{s}{v_1} + \dfrac{s}{v_2}+ .. . . .+ \dfrac{s}{v_n}

    \dfrac{(n)}{V_{av}} = \dfrac{1}{v_1} + \dfrac{1}{v_2}+ .. . . .+ \dfrac{1}{v_n}

    Average velocity is harmonic mean of individual velocities .

    For two equal distance

    V_{av} = \dfrac{2v_1v_2}{v_1+v_2}

  7. Equal Time

    If t_1=t_2 . . . . . =t_n when body covers equal distances then ,

    V_{av}= \dfrac{(s_1 +s_2+ , . . .+ s_n)}{(t_1 + t_2+ .. . . .+ t_n)}

    V_{av}= \dfrac{(v_1t_1 +v_2t_2+ , . . .+ v_nt_n)}{(t_1 + t_2+ .. . . .+ t_n)}

    V_{av}= \dfrac{(v_1 + v_2 + ...+ v_n)}{(n)}

    For two equal time.

    V_{av} = \dfrac{v_1+v_2}{2}

  8. When a moving body covers two equal parts of distances with two different constant velocities v_1 and v_2 in same or any directions, then

    Average speed (v) is given by

    = \dfrac{2v_1v_2}{v_1+v_2}

    But average velocity is given by

    0 ≤ \text{average speed} ≤ \dfrac{2v_1v_2}{v_1+v_2}

    When a moving body travels at two different constant velocities v_1 and v_2 for equal intervals of time, then Average speed =\dfrac{v_1+v_2}{2}

    \dfrac{v_1-v_2}{2} ≤ \text{average speed} ≤ \dfrac{v_1+v_2}{2}

Acceleration

  1. The rate of change of velocity with respect to time is called acceleration. It is a vector quantity.

    \to Acceleration=\dfrac{\text{velocity}}{\text{time taken}}

    \to Average acceleration = \dfrac{\text{change in velocity}}{\text{time taken}}

    \overrightarrow{a_{av}} = \dfrac{(\overrightarrow{v}-\overrightarrow{v_0})}{(t-t_0)} = \dfrac{∆\overrightarrow{v}}{∆t}

    It is defined for a period of time.

    Instantaneous acceleration is limiting the value of average speed as the time interval tends to zero.

    \overrightarrow{a} = \displaystyle{\lim_{∆t \to 0}} \dfrac{∆\overrightarrow{v}}{∆t} = \dfrac{d\overrightarrow{v}}{dt}

    It is defined for a point of time.

  2. Acceleration

    \to If the change in velocity is constant with time. It is either increasing or decreasing by an equal amount in equal interval of time. It is a uniform acceleration.

    \to If the change in velocity is different with time. It is either increasing or decreasing by an unequal amount in equal interval of time. It is a non-uniform acceleration.

  3. If \overrightarrow{s} = \overrightarrow{s}(t)

    \to velocity(v) = \dfrac{d\overrightarrow{s}}{dt}

    \to acceleration(a) = \dfrac{d\overrightarrow{v}}{dt} = \dfrac{d^2\overrightarrow{s}}{dt^2}

    \to s \propto t^0 \to \text{Rest}

    \to s \propto t^1 \to \text{Uniform velocity}

    \to s \propto t^2 \to \text{Uniform acceleration}

    \to s \propto t^3 \to \text{Variable acceleration}

  4. If \overrightarrow{v} = \overrightarrow{v}(t)

    \to displacement(s) = \int \overrightarrow{v} dt

    \to acceleration(a) = \dfrac{d\overrightarrow{v}(t)}{dt}

  5. If \overrightarrow{a} = \overrightarrow{a}(t)

    \to velocity(v) = \int \overrightarrow{a} dt

    \to displacement(s) = \int \overrightarrow{v} dt

Motion with Constant Acceleration

  1. When the motion of the body starts from rest and moves with constant acceleration 'a'. The distance traveled and velocity in nth second is given by:

    s_n = \dfrac{a}{2}(2n-1)

    v_n = at

    \to The distance traveled in successive equal interval of time are in the ratio of 1:3:5:7 ...: (2n-1)

    i.e. s_1:s_2:s_3:...:s_n = 1:3:5:...:(2n-1)

    \to The distance after the end of the successive equal interval of time are in the ratio of 1:4:9:16 ...:n^2

    i.e. s_1:(s_1+s_2):(s_1+s_2+s_3):...(s_1+s_2+...+s_n) = 1:4:9:...:n^2

    \to The velocity after the end of the successive equal interval of time are in the ratio of 1:2:3:4 ...:n

    i.e. v_1:v_2:v_3:...:v_n= 1:2:3:...:n

    \to The time taken to reach the end of successive equal intervals of time are in the ratio of 1:\sqrt{2}:\sqrt{3}:\sqrt{4} ...:\sqrt{n}

    i.e. t_1: (t_1+t_2):(t_1+t_2+t_3):...:(t_1+t_2+t_3+...+t_n)=1:\sqrt{2}:\sqrt{3}:\sqrt{4} ...:\sqrt{n}

    \to The time taken to cover successive equal intervals of time are in the ratio of 1:\sqrt{2}-1:\sqrt{3}-\sqrt{2}:\sqrt{4}-\sqrt{3} ...:\sqrt{n}-\sqrt{n-1}

    i.e. t_1: t_2:t_3:...:t_n=1:\sqrt{2}-1:\sqrt{3}-\sqrt{2}:\sqrt{4}-\sqrt{3} ...:\sqrt{n}-\sqrt{n-1}

  2. When the motion of the body starts from rest and moves with constant acceleration 'a'. If v_1,v_2,v_3,...,v_n be the velocities at successive points at equal separation x, then

    \to v_2^2 -v_1^2 = v_3^2 - v_2^2 = v_4^2 - v_3^2 =...= v_n^2 - v_{n-1}^2= 2ax

  3. When the motion of the body starts from rest and moves with constant acceleration 'a'. If v_1,v_2,v_3,...,v_n be the velocities at equal time interval t, then

    \to v_2 -v_1= v_3- v_2 = v_4- v_3 =...= v_n - v_{n-1}= at

  4. When a particle accelerates from rest for time t_1 over distance x_1 and at the acceleration a_1 and then retards to rest at the rate of a_2 over distance x_2 in time t_2, then

    \to \dfrac{a_1}{a_2} = \dfrac{t_2}{t_1} = \dfrac{x_2}{x_1}

    \to Net acceleration, (a) = \dfrac{a_1a_2}{a_1+a_2}

    \to Maximum velocity (v_{max}) = a_1t_1= a_2t_2 = \sqrt{2a_1x_1} = \sqrt{2a_2x_2} = \sqrt{\dfrac{2a_1a_2(x_2+x_2)}{a_1+a_2}} = \dfrac{a_1a_2}{a_1+a_2} \times \text{total time}

    \to Average velocity v_{av} = \dfrac{a_1a_2}{2(a_1+a_2)} \times (t_1+t_2)

Motion Under Gravity

  1. A body is released from 'H' under the influence of gravity takes time t to reach the ground. Then,

    \to The time taken to reach the ground is t = \sqrt{\dfrac{2H}{g}}

    \to The velocity with which it hits ground is t = \sqrt{2gH} = gt

    \to The distance covered in last second h_t = \dfrac{g}{2}(2t-1)

    \to The velocity of body 'h' height above ground v_h = \sqrt{2g(H-h)}

    \to The average velocity for entire motion v_{av} = \dfrac{gt}{2} = \sqrt{\dfrac{gh}{2}}

    \to The distance traveled in successive equal interval of time are in the ratio of 1:3:5:7 ...: (2n-1)

    \to The distance after the end of the successive equal interval of time are in the ratio of 1:4:9:16 ...:n^2

  2. When a body is thrown vertically upward with initial velocity u under gravity and returns to the thrower's hand. then

    \to Maximum height attached:

    H = \dfrac{u^2}{2g}

    \to Time taken to reach the maximum height

    t= \dfrac{u}{g} = \sqrt{\dfrac{2H}{g}}

    \to Time of flight

    T= 2t = 2 \times \sqrt{\dfrac{2H}{g}} = \sqrt{\dfrac{8H}{g}}

    \to Velocity of the body at height 'h' above the ground

    v_h = \sqrt{2g(H-h)} = u \sqrt{1 - \dfrac{h}{H}}

    \to Average speed for entire trip in upward motion: v_{av} = \dfrac{u}{2} = \sqrt{\dfrac{gH}{2}}

    \to Distance covered in last second of upward motion is: h_t = u- \dfrac{g}{2}(2t-1) = \dfrac{g}{2}

    The body passes a point at height twice after time t_1 and t_2 from starting then

    \to Time to reach maximum height t = \dfrac{t_1+t_2}{2}

    \to Total time of flight T =t_1+t_2

    \to Height of that point from the ground, h = \dfrac{g}{2} t_1t_2

    \to Maximum height attained, H = \dfrac{g}{8} (t_1+t_2)^2

    \to Initial velocity, (u) = \dfrac{g}{2}(t_1+t_2)

  3. When a body of mass 'm' is released from rest from height 'h' along a smooth inclined plane having inclination angle \theta and length 'l', then

    \to Velocity of the body at the bottom of inclined plane v = \sqrt{2gh} = \sqrt{2gl \sin \theta}

    \to Acceleration down the plane a = g \sin \theta

    \to Time taken to fall down the plane t = \sqrt{\dfrac{2h}{g}} \cosec \theta

    \to Time taken to slide down along different smooth inclined planes of different inclination from same height are in ratio t_1:t_2:... = \cosec \theta_1: \cosec \theta_2: ...

    \to Time taken to slide down along different smooth inclined planes of equal length of different inclination are in the ratio t_1:t_2:... = \sqrt{\cosec \theta_1}: \sqrt{\cosec \theta_2}: ...

Relative Velocity

  1. Introduction
  2. Let velocity of body A be \overrightarrow{v_A} and the velocity of body B be \overrightarrow{v_B}.

    \to