A parlicle moves in a straight linc under tho actions a...

A parlicle moves in a straight linc under tho actions a retarding forec. If its initial speed is $u$ and the retarding forec is $m K v^{3}$ where $m$ is its mass, $v$ is velocity at any instant and $K$ is a constant, show that $\frac{1}{v}-\frac{1}{u}-K x, x$ being the distance covered before its speed is reduced to $v .$ Show that the time taken to travel $x$ is given by $t-\frac{K x^{2}}{2}+\frac{x}{u}$ Solution $m \frac{d v}{d t}-m k v^{3} \quad$ or $\frac{d v}{d t}-K v^{3} \quad \Rightarrow \frac{d v}{v^{3}}-K d t$ Intcgrating, $-\frac{1}{2 v^{2}}=-K t+C . \quad$ When $t=0, v=u \quad \therefore \quad-\frac{1}{2 v^{2}}=-K t-\frac{1}{2 u^{2}}$ or $\left(\frac{1}{u^{2}}-\frac{1}{v^{2}}\right)=-2 \mathrm{~K} t \quad \Rightarrow \quad 2 K t=\frac{1}{v^{2}}-\frac{1}{u^{2}}$ Also $m \frac{v d v}{d x}--m K v^{3} \quad \therefore \quad-\frac{d v}{v^{2}}-K d x$ Integrating, $\frac{1}{v}-k x \mid C$ When $x-0, v-u \quad \therefore \quad-\frac{1}{u}+\frac{1}{v}-K x$ or $\frac{1}{v}-\frac{1}{u}-K x$ But $2 \mathrm{~K} \mathrm{l}-\left(\frac{1}{v^{2}} \frac{1}{u^{2}}\right)$ $t=\left(\frac{1}{v^{2}}-\frac{1}{u^{2}}\right) \frac{1}{2 K}=\left(\frac{1}{v}-\frac{1}{u}\right)\left(\frac{1}{v}+\frac{1}{u}\right) \frac{1}{2 K}=K x\left(\frac{2}{u}+K x\right) \frac{1}{2 K} ; \quad \therefore \quad t=\frac{x}{u}+\frac{K x^{2}}{2}$

A parlicle moves in a straight linc under tho actions a retarding forec. If its initial speed is $u$ and the retarding forec is $m K v^{3}$ where $m$ is its mass, $v$ is velocity at any instant and $K$ is a constant, show that $\frac{1}{v}-\frac{1}{u}-K x, x$ being the distance covered before its speed is reduced to $v .$ Show that the time taken to travel $x$ is given by $t-\frac{K x^{2}}{2}+\frac{x}{u}$
Solution $m \frac{d v}{d t}-m k v^{3} \quad$ or
$\frac{d v}{d t}-K v^{3} \quad \Rightarrow \frac{d v}{v^{3}}-K d t$
Intcgrating, $-\frac{1}{2 v^{2}}=-K t+C . \quad$ When $t=0, v=u \quad \therefore \quad-\frac{1}{2 v^{2}}=-K t-\frac{1}{2 u^{2}}$
or $\left(\frac{1}{u^{2}}-\frac{1}{v^{2}}\right)=-2 \mathrm{~K} t \quad \Rightarrow \quad 2 K t=\frac{1}{v^{2}}-\frac{1}{u^{2}}$
Also $m \frac{v d v}{d x}--m K v^{3} \quad \therefore \quad-\frac{d v}{v^{2}}-K d x$
Integrating, $\frac{1}{v}-k x \mid C$
When $x-0, v-u \quad \therefore \quad-\frac{1}{u}+\frac{1}{v}-K x$ or $\frac{1}{v}-\frac{1}{u}-K x$
But $2 \mathrm{~K} \mathrm{l}-\left(\frac{1}{v^{2}} \frac{1}{u^{2}}\right)$
$t=\left(\frac{1}{v^{2}}-\frac{1}{u^{2}}\right) \frac{1}{2 K}=\left(\frac{1}{v}-\frac{1}{u}\right)\left(\frac{1}{v}+\frac{1}{u}\right) \frac{1}{2 K}=K x\left(\frac{2}{u}+K x\right) \frac{1}{2 K} ; \quad \therefore \quad t=\frac{x}{u}+\frac{K x^{2}}{2}$

Key Concepts

Newton's Second Law

Newton's Second Law, which states that the net force acting on an object is equal to the mass of the object multiplied by its acceleration, provides the fundamental starting point for analyzing dynamics. In problems where forces depend on velocity, this law leads to differential equations that describe the motion of the particle.

Non-linear Resistive Forces

Non-linear resistive or damping forces are forces whose magnitude depends on a non-linear function of velocity. These forces, often arising in contexts such as fluid resistance or air drag, require special treatment in the equations of motion since they make the differential equations more complex compared to linear damping scenarios.

Separation of Variables

Separation of variables is a mathematical technique used to solve differential equations by rearranging them so that each variable appears on a different side of the equation. This method is particularly useful in dynamics where acceleration is expressed as a function of velocity, allowing for the integration with respect to velocity on one side and time or displacement on the other.

Integration with Initial Conditions

After separating the variables in a differential equation, integration is performed to find a general solution. Initial conditions are then applied to solve for integration constants, ensuring that the solution accurately models the physical scenario from the given starting state.

Kinematic Relationships

Kinematic relationships in mechanics establish the connections between velocity, acceleration, time, and displacement. Depending on the variable of interest, the differential equations can be manipulated (for example, by expressing acceleration in terms of displacement) and integrated to obtain useful expressions that describe the motion of the system.

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