One dimension

Motion on straight line

•Distance-Actual distance travelled by body. (scalar)

•Displacement-Shortest distance travelled between two points. (vector)

•“Distance can't be zero but displacement may be zero.”

•Distance ≥Displacement

•Speed-It’s the rate of change of distance.(scalar)$speed=\frac{\text{Distance travelled}}{\text{time taken}}$

•Velocity-It’s the rate of change of displacement.(vector)$velocity=\frac{\text{Displacement}}{\text{Time taken}}$

Instantaneous and average velocity

•Instantaneous speed/velocity-It is the speed/velocity at  a exact point of motion.

 Instantaneous speed/velocity=$\frac{d\vec{x}}{dt}$or $\frac{dx}{dt}$

•Average speed/velocity-It is the average of speed/velocity on motion of particle on large interval of time.

Average speed/velocity =$\frac{\text{Total distance or displacement}}{\text{Total time taken}}$

Methodologies

i. When a body covers a distances $s_1,s_2,s_3,......,s_n$at constant velocities $v_1,v_2,v_3,.....,v_n$then,

$Velocity(v_{\text{average}})=\frac{\text{Total displacement}}{\text{Total time taken}}=\frac{s_1+s_2+.....+s_n}{t_1+t_2+.....+t_n}$

=$\frac{s_1+s_2+........+s_n}{\frac{s_1}{v_1}+\frac{s_2}{v_2}+.......+\frac{s_n}{v_n}}$

so, $\frac{s_1}{v_1}+\frac{s_2}{v_2}+.....+\frac{s_n}{v_n}$=$\frac{s_1+s_2+.....+s_n}{v_{\text{average}}}$

If $s_1=s_2=......=s_n$(covers equal distances)

$\frac{n}{v_{\text{average}}}=\frac{1}{v_1}+\frac{1}{v_2}+......+\frac{1}{v_n}$(H.M. between velocities)

ii.When a body travels with V_1, V_2, V₃,....... V_n  for equal interval of time.

$V_{\text{avg}}$     =$\frac{\text{Total displacement}}{\text{Total time taken}}$=$\frac{s_1+s_2+s_3+.....+s_n}{t_1+t_2+t_3+....+t_n}$

= $\frac{v_1.t_1+v_2.t_2+.....+v_n.t_n}{t_1+t_2+...+t_n}$

 for t₁ = t₂ = t₃ =....... = t_n ( in equal time) then,

$v_{\text{avg}}=\frac{v_1+v_2+....+v_n}{n}$ (A.M. between velocities)

Equations of motion (air resistance is neglected)

If x=displacement, v=velocity, a=acceleration, t=time taken then

$v=\frac{\Delta x}{\Delta t}$( for instantaneous , $\frac{dx}{dt}$)..........(i)

$a=\frac{\Delta v}{\Delta t}$(for instantaneous, $\frac{d^{2}x}{d{t}^2}$)............(ii)

If v=final velocity, u=initial velocity, s=distance travelled, t=time taken, a=constant acceleration then

i. $v=u+at${From (ii), ${\int }_u^{v}dv={\int }_0^{t}a.dt$}...........(iii)

ii.$s=ut+\frac{1}{2}a{t}^2${From (i) and (ii), ${\int }_0^{s}dx={\int }_0^{t}v.dt={\int }_0^t(u+at)dt$ }

iii.$v^2=u^2+2as$ {From (i) $v=\frac{dx}{dt}$ from(ii) $dt=\frac{dv}{a}$ so $v.dv=a.dx={\int }_u^{v}v.dv={\int }_0^{t}a.dt$ }

iv. The distance travelled on $n_{\text{th}}$successive second =$u+\frac{a}{2}(2n-1)$

If air resistance is not neglected time of descent>time of ascent.

Graph of equations of motion

If d=distance, t=time, v=velocity, a=acceleration.

(1) Slope of d-t graph gives avg. velocity.(Intercept on y axis gives initial displacement)

(2) Slope of v-t graph gives avg. acceleration.

(Intercept on y axis gives initial velocity)

(3) Area of v-t graph gives distance covered.

(4) Area of a-t graph gives change in velocity.

(Intercept gives in initial acceleration)