Question

If the velocity of a particle is defined by $\vec{r}-(2 t+1) \hat{i}+3 \hat{j}$ and its position vector at $t=\mathrm{ls}$ is $\vec{r}-4 \hat{i}+3 \hat{j}$, determine the path of the particle.

   If the velocity of a particle is defined by $\vec{r}-(2 t+1) \hat{i}+3 \hat{j}$ and its position vector at $t=\mathrm{ls}$ is $\vec{r}-4 \hat{i}+3 \hat{j}$, determine the path of the particle.

Practice Problem in Physics for the JEE Main and Advanced

Practice Problem in Physics for the JEE Main and Advanced

Abhay Kumar 2015 Edition

Chapter 3, Problem 10

Instant Answer

Solved by Expert Hast Aggarwal

01/15/2022

Step 1

We can separate this into two equations for the x and y components of velocity: $v_x = \frac{dx}{dt} = 2t + 1$ and $v_y = \frac{dy}{dt} = 3$. Show more…

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If the velocity of a particle is defined by $\vec{r}-(2 t+1) \hat{i}+3 \hat{j}$ and its position vector at $t=\mathrm{ls}$ is $\vec{r}-4 \hat{i}+3 \hat{j}$, determine the path of the particle.

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Key Concepts

Velocity Vector

The velocity vector represents the rate at which the position of a particle changes with time, providing both magnitude (speed) and direction. In kinematics, it is expressed as a vector quantity and is fundamental for understanding how the particle moves in space.

Integration in Kinematics

Integration is the mathematical process used to derive the position function from the velocity function. By integrating the velocity components with respect to time, one obtains expressions for the position coordinates that describe the motion of the particle.

Initial Conditions

Initial conditions are the specific values of the position (and sometimes velocity) at a particular time that allow for the determination of the constant of integration. They are essential for producing a unique solution that accurately reflects the particle’s trajectory.

Trajectory of a Particle

The trajectory or path of a particle is the route followed through space, often represented by a relationship between spatial coordinates. By eliminating the time variable from the parametric equations obtained through integration, one can derive a Cartesian equation that directly relates the coordinates, depicting the particle’s path.

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