A remote-controlled car is moving in a vacant parking lot. The velocity of the car as a function of time is given by $\vec{v}\) = $\left$[ 5.00~$\mathrm{m/s}$ - (0.0180~$\mathrm{m/s^3}$)t^2 $\right$] $\hat{i}$ + $\left$[ 2.00~$\mathrm{m/s}$ + (0.550~$\mathrm{m/s^2}$)t $\right$] \(\hat{j}$ . What are $a_{x} (t)$ and $a_{y} (t)$ , the $x$ - and $y$ - components of the car's velocity as functions of time?

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Identify the given velocity function of the car: $ \mathbf{v}(t) = [5.00 \text{ m/s} - (0.0180 \text{ m/s}^3)t^2]\hat{i} + [2.00 \text{ m/s} + (0.550 \text{ m/s}^2)t]\hat{j} $.

Understand that the velocity function is composed of two components: $ v_x(t) = 5.00 \text{ m/s} - (0.0180 \text{ m/s}^3)t^2 $ and $ v_y(t) = 2.00 \text{ m/s} + (0.550 \text{ m/s}^2)t $.

To find the acceleration components, differentiate the velocity components with respect to time. The acceleration is the derivative of velocity.

Calculate the x-component of acceleration: $ a_x(t) = \frac{d}{dt}[5.00 \text{ m/s} - (0.0180 \text{ m/s}^3)t^2] $.

Calculate the y-component of acceleration: $ a_y(t) = \frac{d}{dt}[2.00 \text{ m/s} + (0.550 \text{ m/s}^2)t] $.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Velocity as a Function of Time

Velocity as a function of time describes how the speed and direction of an object change over time. In this problem, the velocity is given as a vector with components in the x and y directions, each expressed as a function of time. Understanding this concept is crucial for determining how the car's motion evolves in both directions.

Differentiation

Differentiation is a mathematical process used to find the rate at which a quantity changes. In physics, it is often used to derive acceleration from velocity. For this problem, differentiating the velocity functions with respect to time will yield the acceleration components ax(t) and ay(t), which describe how the car's velocity changes in the x and y directions.

Vector Components

Vector components break down a vector into parts that align with the coordinate axes, typically x and y. Each component represents the influence of the vector in that direction. In this problem, the velocity vector is split into x and y components, allowing us to analyze the car's motion separately along each axis, which is essential for calculating ax(t) and ay(t).