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?

$$H=\dfrac{u²}{2g}$$