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. (b) At t = 5.00 s, how far is the squirrel from its initial position?

Verified step by step guidance

1. Identify the position function of the squirrel. The position vector is given by \( r = [(0.280 \, \text{m/s})t + (0.0360 \, \text{m/s}^2)t^2] \hat{i} + (0.0190 \, \text{m/s}^3)t^3 \hat{j} \).
2. Substitute the given time \( t = 5.00 \, \text{s} \) into the position function to find the position of the squirrel at that time. Calculate \( r(5.00) \) by plugging in \( t = 5.00 \, \text{s} \) into each component of the vector.
3. Calculate the initial position of the squirrel at \( t = 0 \, \text{s} \) by substituting \( t = 0 \) into the position function, resulting in \( r(0) \).
4. Determine the displacement vector \( \Delta r \) by subtracting the initial position vector \( r(0) \) from the position vector at \( t = 5.00 \, \text{s} \), i.e., \( \Delta r = r(5.00) - r(0) \).
5. Calculate the magnitude of the displacement vector \( \Delta r \) to find how far the squirrel is from its initial position. Use the formula for the magnitude of a vector: \( |\Delta r| = \sqrt{(\Delta x)^2 + (\Delta y)^2} \), where \( \Delta x \) and \( \Delta y \) are the components of the displacement vector.

Key Concepts

Position Vector

A position vector describes the location of an object in space relative to a reference point, typically the origin of a coordinate system. In this case, the position of the squirrel is given as a function of time, incorporating both linear and nonlinear components. The vector notation indicates movement in two dimensions, represented by the unit vectors î and ĵ.

Kinematics

Kinematics is the branch of mechanics that deals with the motion of objects without considering the forces that cause the motion. It involves analyzing position, velocity, and acceleration as functions of time. In this problem, the position function of the squirrel allows us to determine its displacement over a specified time interval.

Displacement

Displacement is a vector quantity that represents the change in position of an object. It is calculated as the final position minus the initial position. In this scenario, to find how far the squirrel is from its initial position at t = 5.00 s, we evaluate the position vector at that time and compare it to the position at t = 0.