Question:
The position x of a particle varies with time, (t) as $x = at^2 - bt^3.$ The acceleration will be zero at time t is equal to
The position x of a particle varies with time, (t) as $x = at^2 - bt^3.$ The acceleration will be zero at time t is equal to
Updated On: May 2, 2024
- $\frac{a}{3b}$
- zero
- $\frac{2a}{3b}$
- $\frac{a}{b}$
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The Correct Option is A
Solution and Explanation
Distance $ (x) = at^2- bt^3.$
Therefore velocity $ (v) = \frac{dx}{dt} = \frac{d}{dt} (at^2- bt^3)$
$=2at -3bt^2$ and
acceleration $ = \frac{dv}{dt} = \frac{d}{dt} (2at- 3bt^2) = 2a - 6bt = 0$
or $\, \, t = \frac{2a}{6b} = \frac{a}{3b}.$
Therefore velocity $ (v) = \frac{dx}{dt} = \frac{d}{dt} (at^2- bt^3)$
$=2at -3bt^2$ and
acceleration $ = \frac{dv}{dt} = \frac{d}{dt} (2at- 3bt^2) = 2a - 6bt = 0$
or $\, \, t = \frac{2a}{6b} = \frac{a}{3b}.$
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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:
- 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.
- 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.