A particle is moving in a circle of radius R . Λt t=0 its speed is v0 and during its motion spee

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A particle is moving in a circle of radius R . Λt t=0 its...

A particle is moving in a circle of radius $R . \Lambda \mathrm{t} t=0$ its speed is $v_{0}$ and during its motion speed varies as $\frac{d v}{d t}-\frac{-k v^{2}}{R}$ where $k$ is a positive constant. Find the speed of the particle: (a) after $1 \mathrm{sec}$; (b) after 1 revolution.

A particle is moving in a circle of radius $R . \Lambda \mathrm{t} t=0$ its speed is $v_{0}$ and during its motion speed
varies as $\frac{d v}{d t}-\frac{-k v^{2}}{R}$ where $k$ is a positive constant. Find the speed of the particle: (a) after $1 \mathrm{sec}$; (b) after 1 revolution.

Key Concepts

Separable Differential Equations

Separable differential equations are a class of ordinary differential equations that can be rearranged so that all terms involving one variable appear on one side of the equation and all terms involving the other variable on the opposite side. This technique is useful in kinematics for solving equations where the rate of change of a variable (such as speed) is expressed in terms of the variable itself. By integrating both sides, one can typically find an explicit formula that describes the variable as a function of time.

Integration in Kinematics

Integration in kinematics is used to relate quantities such as velocity, acceleration, and displacement. When a particle’s speed or acceleration is given by a function of time (or even as a function of speed), integration provides a method to determine how far the particle has traveled or to find the time required to cover a given distance. This is particularly relevant in non-uniform motion, where the relationship between distance and time is not linear.

Circular Motion

Circular motion refers to the movement of a particle along a circular path. In physics, this concept is used to analyze the position, velocity, and acceleration of an object moving in a circle, taking into account both radial and tangential components. The radius of the circle plays a critical role in determining the geometric relationships, such as the arc length traveled during motion and the centripetal acceleration that acts perpendicular to the velocity vector.

Non-uniform Circular Motion

Non-uniform circular motion describes the case when a particle’s speed along the circular path is not constant. In this scenario, there is a tangential acceleration in addition to the centripetal (radial) acceleration. The change in speed requires an analysis that couples the dynamics of the motion with the constraints of the circular path. Such problems often lead to differential equations that describe how the speed evolves with time.

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