Question:
The displacement x of a particle varies with time t as $x = ae^{-\alpha t}+be^{\beta t}$ where a, b, $\alpha$ and $\beta$ are positive constants. The velocity of the particle will
The displacement x of a particle varies with time t as $x = ae^{-\alpha t}+be^{\beta t}$ where a, b, $\alpha$ and $\beta$ are positive constants. The velocity of the particle will
Updated On: Jun 9, 2024
- be independent of $\beta$
- drop to zero when $\alpha = \beta $
- go on decreasing with time
- go on increasing with time
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The Correct Option is D
Solution and Explanation
$x = ae^{-\alpha t}+be^{\beta t}$
$\frac{dx}{dt} = -a \alpha e^{-\alpha t}+b \beta e^{\beta t}$
$v = -a \alpha e^{-\alpha t}+b \beta e^{\beta t}$
For certain value of t, velocity will increases.
$\frac{dx}{dt} = -a \alpha e^{-\alpha t}+b \beta e^{\beta t}$
$v = -a \alpha e^{-\alpha t}+b \beta e^{\beta t}$
For certain value of t, velocity will increases.
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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.