A particle moves in the x y plane with only an x -component...

A particle moves in the $x y$ plane with only an $x$ -component of acceleration of $2 \mathrm{~ms}^{-2}$. The particle starts from the origin at $t=0$ with an initial velocity having an $x$ -component of $8 \mathrm{~ms}^{-1}$ and $y$ -component of $-15 \mathrm{~ms}^{-1}$. The total velocity vector at any time $t$ is a. $[(8+2 t) \hat{i}-15 \hat{j}] \mathrm{ms}^{-1}$ b. zero c. $2 t \hat{i}+15 \hat{j}$ d. directed along $z$ -axis

A particle moves in the $x y$ plane with only an $x$ -component of acceleration of $2 \mathrm{~ms}^{-2}$. The particle starts from the origin at $t=0$ with an initial velocity having an $x$ -component of $8 \mathrm{~ms}^{-1}$ and $y$ -component of $-15 \mathrm{~ms}^{-1}$. The total velocity vector at any time $t$ is
a. $[(8+2 t) \hat{i}-15 \hat{j}] \mathrm{ms}^{-1}$
b. zero
c. $2 t \hat{i}+15 \hat{j}$
d. directed along $z$ -axis

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Key Concepts

Kinematics in Two Dimensions

This concept involves analyzing the motion of particles in a plane by resolving vectors into their x and y components. It is based on the idea that the equations of motion for each component can be applied independently when acceleration is constant, allowing the overall motion to be determined by combining the independent motions along each axis.

Constant Acceleration Equations

These equations, such as v = u + at, are fundamental in predicting a particle's velocity when it experiences constant acceleration. They show how the initial velocity and the constant acceleration affect the motion over time, making it possible to calculate the updated velocity vector for each independent direction in problems involving uniform acceleration.

Vector Component Analysis

This key concept involves breaking down vectors into their perpendicular components (typically x and y directions) to simplify the analysis of motion. By treating each component separately, one can use basic kinematic equations to determine the contribution of each direction to the overall motion, and then reconstruct the total vector through vector addition.

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