Question

The displacement \[x\] of a particle at a time $t$ is given by $x = A{t^2} + Bt + C$ where $A$, $B$ and $C$ are constants and $v$ is the velocity of a particle, then the value $4Ax - {v^2}$ is
$1)4AC + {B^2}$
$2)4AC - {B^2}$
$3)2AC - {B^2}$
$4)2AC + {B^2}$

Answer 1

Hint: We are given an equation for the displacement of a particle at a given time. The equation consists of three constants and with a certain velocity and we are required to find the value of the other equation.

Complete step by step solution:
We know that the rate of change of displacement is called the velocity of the particle. The rate of change of a quantity with respect to time is given by differentiating the quantity with respect to time. So we will differentiate the given equation with respect to time and see if we can find the value of the asked equation.
Given $x = A{t^2} + Bt + C$
Where $x$ is the displacement
Differentiating the given equation we get
$v = 2At + B$
Where $v$ is the velocity
Squaring both sides of the equation we get
$ \Rightarrow {v^2} = 4{A^2}{t^2} + 4ABt + {B^2} - - - (1)$
Multiplying the equation by $4A$ we get
$ \Rightarrow 4Ax = 4{A^2}{t^2} + 4ABt + 4AC - - - (2)$
Subtracting equation 2 from equation 1, we get
$ \Rightarrow {v^2} - 4Ax = {B^2} - 4AC$
Rearranging the equation we get
$ \Rightarrow 4Ax - {v^2} = 4AC - {B^2}$
Hence option 2 is correct.

Additional Information: The zeroth derivative of displacement is itself, displacement, the first derivative of displacement is velocity and the second derivative of displacement is acceleration. Displacement is the measurement of the length of the shortest distance from the initial point to the final point. It is a vector quantity having direction as well as magnitude. Distance is a scalar quantity with no direction and only magnitude.

Note: If we differentiate the velocity further we will get the acceleration of the particle. Acceleration is the rate of change of velocity with respect to time. The acceleration of the particle is positive when the velocity increases with time.