The sprinter begins from rest at position A and accelerates...

The sprinter begins from rest at position $A$ and accelerates along the track. If the stationary tracking camera at $O$ is rotating counterclockwise at the rate of 12.5 deg/s when the sprinter passes the $60-\mathrm{m}$ mark, determine the speed $v$ of the sprinter and the value of $\dot{r}$

Key Concepts

Relationship Between Angular and Linear Motion

The connection between angular and linear motion arises through the radial distance: the linear speed along a path that is a function of angle is determined by multiplying the angular speed by the radius. This relationship is fundamental in converting rotational measurements into linear velocities.

Differentiation in Polar Coordinates

Differentiation in polar coordinates involves expressing the rates of change of a point’s position in terms of its changing radial distance and angular position. The chain rule is used to differentiate these components when calculating quantities like the time rate of change of the distance (radial velocity) and the speed of an object moving in a curved path.

Angular Velocity

Angular velocity describes how fast an object rotates or the rate at which an angle changes. In many problems, it is essential to convert angular speed between units, such as from degrees per second to radians per second, because standard physical equations involving rotations require radian measure.

Polar Coordinates

Polar coordinates provide a framework for describing the position of a point in a plane using a distance from a fixed origin and an angle relative to a reference direction. They are particularly useful in problems where motion is naturally expressed in terms of radial and angular components.

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