The position of a squirrel running in a park is given by r = [(0.280 m/s)t + (0.0360 m/s2)t2]î + (0.0190 m/s3)t3ĵ. (a) What are υx(t) and υy(t), the x- and y-components of the velocity of the squirrel, as functions of time?

Verified step by step guidance

1. Identify the position vector components in the x and y directions. For the x-direction, the position component is given by $r_x(t) = 0.280t + 0.0360t^2$. For the y-direction, the position component is $r_y(t) = 0.0190t^3$.
2. Recall that the velocity components can be found by differentiating the position components with respect to time. Use the formula $v(t) = \frac{dr(t)}{dt}$ to find the velocity components.
3. Differentiate the x-component of the position, $r_x(t) = 0.280t + 0.0360t^2$, with respect to time to find the x-component of the velocity, $v_x(t)$. Apply the power rule of differentiation: $\frac{d}{dt}(at + bt^2) = a + 2bt$.
4. Differentiate the y-component of the position, $r_y(t) = 0.0190t^3$, with respect to time to find the y-component of the velocity, $v_y(t)$. Use the power rule: $\frac{d}{dt}(ct^3) = 3ct^2$.
5. Write the final expressions for $v_x(t)$ and $v_y(t)$ using the results from the differentiation. These expressions represent the x- and y-components of the velocity of the squirrel as functions of time.

Key Concepts

Position Vector

The position vector describes the location of an object in space as a function of time. In this case, the position of the squirrel is given in terms of its x and y coordinates, represented by the equation r(t). Understanding how to interpret and manipulate this vector is essential for deriving other quantities like velocity.

Velocity Components

Velocity is defined as the rate of change of position with respect to time. For a two-dimensional motion, the velocity can be broken down into its x- and y-components, υx and υy. These components can be found by differentiating the position vector with respect to time, providing insight into the squirrel's motion in each direction.

Differentiation

Differentiation is a fundamental concept in calculus that involves finding the derivative of a function. In the context of this problem, differentiating the position vector with respect to time allows us to calculate the velocity components. This process is crucial for understanding how the position of the squirrel changes over time and is a key step in solving the question.