Question:

The relation between time $t$ and distance $x$ is $t = ax^2 + bx$, where $a$ and $b$ are constants. The acceleration is:

Updated On: Mar 26, 2024
  • $-2abv^2$
  • $2bv^3$
  • $-2av^3$
  • $2av^2$
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The Correct Option is C

Solution and Explanation

Given $t = ax^2 + bx$ Differentiating $w.r.t. \,t$ $\frac{dt}{dt}=2\,ax \frac{dt}{dt}+b \frac{dx}{dt}$ $v=\frac{dx}{dt}=\frac{1}{\left(2\,ax+b\right)}$ Again differentiating, $w.r.t. \,t$ $\frac{d^{2}x}{dt^{2}}=\frac{-2a}{\left(2\,ax+b\right)^{2}}.\frac{dx}{dt}$ $\therefore f=\frac{d^{2}x}{dt^{2}}$ $=-\frac{-1}{\left(2\,ax+b\right)^{2}}. \frac{2a}{\left(2\,ax+b\right)}$ or $f=\frac{-2a}{\left(2\,ax+b\right)^{3}}$ $\therefore f=-2\,av^{3}$
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Concepts Used:

Motion in a straight line

The motion in a straight line is an object changes its position with respect to its surroundings with time, then it is called in motion. It is a change in the position of an object over time. It is nothing but linear motion. 

Types of Linear Motion:

Linear motion is also known as the Rectilinear Motion which are of two types:

  1. Uniform linear motion with constant velocity or zero acceleration: If a body travels in a straight line by covering an equal amount of distance in an equal interval of time then it is said to have uniform motion.
  2. Non-Uniform linear motion with variable velocity or non-zero acceleration: Not like the uniform acceleration, the body is said to have a non-uniform motion when the velocity of a body changes by unequal amounts in equal intervals of time. The rate of change of its velocity changes at different points of time during its movement.