A web page designer creates an animation in which a dot on a...

A web page designer creates an animation in which a dot on a computer screen has position $$ \vec{r}=\left[4.0 \mathrm{~cm}+\left(2.5 \mathrm{~cm} / \mathrm{s}^{2}\right) t^{2}\right] \hat{\imath}+(5.0 \mathrm{~cm} / \mathrm{s}) t \hat{\jmath} $$ (a) Find the magnitude and direction of the dot's average velocity between $t=0$ and $t=2.0 \mathrm{~s}$. (b) Find the magnitude and direction of the instantaneous velocity at $t=0, t=1.0 \mathrm{~s},$ and $t=2.0 \mathrm{~s}$. (c) Sketch the dot's trajectory from $t=0$ to $t=2.0 \mathrm{~s},$ and show the velocities calculated in part (b).

A web page designer creates an animation in which a dot on a computer screen has position $$ \vec{r}=\left[4.0 \mathrm{~cm}+\left(2.5 \mathrm{~cm} / \mathrm{s}^{2}\right) t^{2}\right] \hat{\imath}+(5.0 \mathrm{~cm} / \mathrm{s}) t \hat{\jmath} $$
(a) Find the magnitude and direction of the dot's average velocity between $t=0$ and $t=2.0 \mathrm{~s}$. (b) Find the magnitude and direction of the instantaneous velocity at $t=0, t=1.0 \mathrm{~s},$ and $t=2.0 \mathrm{~s}$. (c) Sketch the dot's trajectory from $t=0$ to $t=2.0 \mathrm{~s},$ and show the velocities calculated in part (b).

Key Concepts

Position Vector

A position vector specifies the location of an object in space relative to an origin, using a coordinate system. It expresses the object's position as a combination of its components along mutually perpendicular axes, which provides a basis for further analysis of its motion.

Average Velocity

Average velocity is defined as the displacement of an object divided by the total time taken. It is a vector quantity that provides both the magnitude (speed) and direction of the overall change in position during an interval, regardless of the detailed path taken.

Instantaneous Velocity

Instantaneous velocity is the velocity of an object at a specific moment in time. It is obtained by taking the derivative of the position vector with respect to time, yielding a vector that indicates both the speed and direction of motion at that exact moment.

Differentiation in Kinematics

Differentiation is a mathematical tool used in kinematics to transition from position to velocity by computing the time derivative of a position function. This operation isolates the rate of change of position, providing insights into the object's instantaneous motion characteristics.

Magnitude and Direction Calculation

Calculating the magnitude and direction of a vector involves determining the scalar length (using methods such as the Pythagorean theorem in a two-dimensional space) and the angle it makes relative to a reference axis. This process is essential for fully describing vector quantities like velocity.

Trajectory Sketching

Trajectory sketching involves drawing the path that an object follows as it moves, based on its position function. This graphical representation helps visualize the motion and the changes in velocity and direction over time, giving a complete picture of the object's kinematics.

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