Question

write the mathematical expression for position time equation and also mention their meaning ​

Answer 1

Mathematical expression for position time equation:

Derivation:

since, we know that -

$velocity=\dfrac{distance}{time} \\\\v(t)=\dfrac{dx}{dt}$

from 1st equation of motion , we have

$\dfrac{dx}{dt}=v(t)=u+at$

On integration ,

$x(t)=\int {dx} =\int {u+at} \, dt \\x(t)=u(t) + a(t^{2}/2) +D\\=ut+\dfrac{1}{2}at^{2}+D$..............(1)

Here, D is any arbitrary constant , which we need to determine.

therefore, let position of the object at t=0 is given, say $x_{0}$ .

Then at t=0 equation (1) becomes;

$x_{0}=0+0+D\\x_{0}=D$

Putting value of D in equation (1) , we get-

$x(t)=\dfrac{1}{2} at^{2} +ut+ x_{0}$

where, x(t) is the position of the object as function of time,

a is the acceleration, t is the time ,

u is the initial velocity and $x_{0}$ is the initial position of the object.