The x and y coordinates of a particle at any time t are...

The $x$ and $y$ coordinates of a particle at any time $t$ are given by $x=7 t+4 t^{2}$ and $y=5 t$, where $x$ and $y$ are in $\mathrm{m}$ and $t$ in s. The acceleration of the particle at $5 \mathrm{~s}$ is a. zero b. $8 \mathrm{~m} / \mathrm{s}^{2}$ c. $20 \mathrm{~m} / \mathrm{s}^{2}$ d. $40 \mathrm{~m} / \mathrm{s}^{2}$

The $x$ and $y$ coordinates of a particle at any time $t$ are given by $x=7 t+4 t^{2}$ and $y=5 t$, where $x$ and $y$ are in $\mathrm{m}$ and $t$ in
s. The acceleration of the particle at $5 \mathrm{~s}$ is
a. zero
b. $8 \mathrm{~m} / \mathrm{s}^{2}$
c. $20 \mathrm{~m} / \mathrm{s}^{2}$
d. $40 \mathrm{~m} / \mathrm{s}^{2}$

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Key Concepts

Parametric Equations in Kinematics

This concept involves describing the motion of a particle using separate equations for each coordinate as functions of time. It allows analysis of motion in the x and y directions independently, which is especially useful in two-dimensional projectile or curvilinear motion scenarios.

Differentiation to Find Velocity and Acceleration

Differentiation is used in kinematics to determine velocity and acceleration from position functions. By taking the first derivative of the position with respect to time, we obtain the velocity components, and by taking the second derivative, we obtain the acceleration components, which describe how the velocity of the particle is changing over time.

Acceleration in Two-Dimensional Motion

Acceleration is a vector quantity that represents the rate of change of velocity in each component of motion. In two-dimensional motion, once the acceleration components are determined through differentiation, the overall acceleration can be analyzed by considering each independent contribution from the x and y directions.

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