A web page designer creates an animation in which a dot on a computer screen has position r→=[4.0cm⁡+(2.5 cm/s2)t2]i+(5.0cm/s⁡)tj\overrightarrow{r}=\left\lbrack4.0\operatorname{cm}+\left(2.5\text{ cm/s}^2\$$right)t^2$$\right\rbrack\mathbf{i}+\left(5.0\operatorname{\text{cm/s}}\right)t\mathbf{j}. Find the magnitude and direction of the dot's average velocity between t=0t = 0 and t=2.0st=2.0s.

Ch 03: Motion in Two or Three Dimensions

Young & Freedman Calc - University Physics 14th Edition

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Young & Freedman Calc 14th Edition

Ch 03: Motion in Two or Three Dimensions

문제 3a

3장, 문제 3a

A web page designer creates an animation in which a dot on a computer screen has position $\vec{r} = [4.0 cm + (2.5 {cm/s}^{2}) t^{2}] i + (5.0 cm/s) t j$ . Find the magnitude and direction of the dot's average velocity between $t equals 0$ and $t equals 2.0 s$ .

검증된 단계별 안내

Identify the position vector $ \mathbf{r}(t) = [4.0\, \text{cm} + (2.5\, \text{cm/s}^2)t^2]\mathbf{i} + (5.0\, \text{cm/s})t\mathbf{j} $.

Calculate the position vector at $ t = 0 $ by substituting $ t = 0 $ into $ \mathbf{r}(t) $.

Calculate the position vector at $ t = 2.0 $ s by substituting $ t = 2.0 $ s into $ \mathbf{r}(t) $.

Determine the change in position vector $ \Delta \mathbf{r} = \mathbf{r}(2.0) - \mathbf{r}(0) $.

Calculate the average velocity $ \mathbf{v}_{\text{avg}} = \frac{\Delta \mathbf{r}}{\Delta t} $ and find its magnitude and direction using $ \text{magnitude} = \sqrt{v_{x}^2 + v_{y}^2} $ and $ \text{direction} = \tan^{-1}\left(\frac{v_{y}}{v_{x}}\right) $.

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주요 개념

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

Average Velocity

Average velocity is defined as the total displacement divided by the total time taken. It is a vector quantity, meaning it has both magnitude and direction. To find the average velocity, calculate the change in position over the given time interval and divide by the duration of that interval.

Vector Components

Vectors have components that describe their influence in different directions, typically represented by î, ĵ, and k̂ for the x, y, and z axes respectively. In this problem, the position vector r→ is given in terms of its î and ĵ components, which need to be analyzed separately to determine the overall displacement and velocity.

Magnitude and Direction of a Vector

The magnitude of a vector is its length, calculated using the Pythagorean theorem for its components. The direction is given by the angle it makes with a reference axis, often found using trigonometric functions like tangent. For the average velocity, compute the magnitude from its components and use inverse tangent to find the direction.