Motion in One Dimension

Revision for basic formulae that will be useful in JEE/IAT or Boards.

What is the formula for \overrightarrow{v_{avg}} in terms of displacement?

\dfrac{\text{total displacement}}{\text{total time taken}}=\dfrac{\Delta\overrightarrow{r}}{\Delta t}=\dfrac{\overrightarrow{r_2}-\overrightarrow{r_1}}{t_2-t_1}

What is the formula for v_{avg} when given v as a function of t?

v_{avg}=\dfrac{\int\limits_{t_1}^{t_2}\ v\ dt}{t_2-t_1}

What is the formula for \overrightarrow{a_{avg}} in terms of change in velocity?

\dfrac{\text{change in velocity}}{\text{total time taken}}=\dfrac{\Delta\overrightarrow{v}}{\Delta t}=\dfrac{\overrightarrow{v_2}-\overrightarrow{v_1}}{t_2-t_1}

What is the formula for a_{avg} when given a as a function of t?

a_{avg}=\dfrac{\int\limits_{t_1}^{t_2}\ a\ dt}{t_2-t_1} 

When given displacement as a function of time (x=f(t)), how can you find velocity?

v=\dfrac{dx}{dt}

When given displacement as a function of time (x=f(t)), how can you find acceleraton?

a=\dfrac{d²y}{dx²}

When given velocity as a function of time (v=f(t)), how can you find acceleration?

a=\dfrac{dv}{dx}

When given velocity as a function of displacement (v=f(x)), how can you find acceleration?

a=v\dfrac{dv}{dx} 

When given velocity as a function of time (v=f(t)), how can you find an equation of displacement as a function of time?

v=f(t)=\dfrac{dx}{dt}

\implies \int dx=\int f(t)dt=x   

When given velocity as a function of displacement (v=f(x)), how can you find an equation for displacement as a function of time?

v=f(x)=\dfrac{dx}{dt}

\implies \int \dfrac{dx}{f(x)}=\int dt   and solve for x.

When given acceleleration as a function of time (a=f(t)), how can you get velocity?

a=\dfrac{dv}{dt}

When given acceleleration as a function of displacement (a=f(x)), how can you get velocity?

a=v\dfrac{dv}{dx}

When given acceleleration as a function of velocity (a=f(v)), how can you get velocity as a function of time?

f(v)=\dfrac{dv}{dt}

Integrate then solve.

When given acceleleration as a function of velocity (a=f(v)), how can you get velocity as a function of displacement?

f(v)=v\dfrac{dv}{dx}

Integrate then solve

What is the formula for displacement in n^{th} second when given initial velocity and uniform acceleration?

S_n=u+\dfrac{a}{2}(2n-1)

For motion under gravity, what is the formula for time of ascent, or time of descent? (two formulae)

t=\dfrac{u}g=\sqrt{\dfrac{2H}{g}}   

For motion under gravity, what is the total time of flight, when projected from ground?

T=\dfrac{2u}{g}  

For motion under gravity, what is the formula for maximum height reached?

H=\dfrac{u²}{2g}