Chapter 3: Motion in a Circle, Projectile Motion, and Relative Velocity

Motion in a Circle

Uniform Circular Motion

• Definition: A car moving along a circular path with constant speed. The acceleration is always directed toward the center of the circular path.

• Acceleration Direction: Exactly perpendicular to the velocity vector; there is no parallel component of acceleration in uniform circular motion.

• Effect of Acceleration: Only changes the car's direction, not its speed.

• Characteristics (Fig. 3.27a):

  • Acceleration ($a_{rad}$) has constant magnitude but varying direction.

  • Velocity ($\vec{v}$) and acceleration ($\vec{a}$) are always perpendicular.

  • Instantaneous acceleration always points toward the center of the circle, hence called centripetal acceleration.

• Magnitude of Centripetal Acceleration:

  • $a = \frac{v^2}{R}$

• Period (T):

  • The time for one complete revolution.

  • The speed can also be expressed as $v = \frac{2\pi R}{T}$, where $R$ is the radius of the circular path.

  • Therefore, the magnitude of centripetal acceleration can also be expressed in terms of the period:

    $a = \frac{4\pi^2 R}{T^2}$

Nonuniform Circular Motion

• Definition: Occurs if the speed of the object varies while it moves along a circular path.

• Acceleration Components:

  • Radial acceleration component ($a_{rad})$): Still present and directed toward the center, calculated as $a_{rad} = \frac{v^2}{R}$. This component changes the direction of the velocity.

  • Tangential acceleration component ($a_{tan}$): Also present, parallel to the instantaneous velocity. This component changes the speed of the object.

• Visual Representation (Fig. 3.27b & 3.27c):

  • Speeding Up: The tangential acceleration component is in the same direction as the velocity.

  • Slowing Down: The tangential acceleration component is in the opposite direction to the velocity.

Examples of Centripetal Acceleration

• EXAMPLE 3.11: Centripetal acceleration on a curved road (Aston Martin V12 Vantage)

  • Scenario: Sports car with a maximum lateral acceleration (centripetal acceleration) of $0.97g$ without skidding.

  • Given: Max $a_{rad} = 0.97g = (0.97)(9.8 \text{ m/s}^2) = 9.5 \text{ m/s}^2$. Constant speed $v = 40 \text{ m/s}$.

  • Task: Find the radius $R$ of the tightest unbanked curve it can negotiate.

  • Method: Use the formula $a_{rad} = \frac{v^2}{R}$ to solve for $R = \frac{v^2}{a_{rad}}$.

• EXAMPLE 3.12: Centripetal acceleration on a carnival ride

  • Scenario: Passengers move at constant speed in a horizontal circle.

  • Given: Radius $R = 5.0 \text{ m}$. Completes a circle in $T = 4.0 \text{ s}$.

  • Task: What is their acceleration?

  • Method: Use the formula $a_{rad} = \frac{4\pi^2 R}{T^2}$.

Projectile Motion (Comparison to Circular Motion)

• Key Distinction: In projectile motion, velocity ($\vec{v}$) and acceleration ($\vec{a}$) are perpendicular only at the very peak of the trajectory.

• Acceleration in Projectile Motion: The acceleration due to gravity ($\vec{g}$) is constant in both magnitude ($9.8 \text{ m/s}^2$) and direction (always downward), unlike in circular motion where acceleration's direction continuously changes.

Relative Velocity

Definition and Frames of Reference

• Relative Velocity: The velocity of a moving object as observed by a particular observer (or relative to a specific reference frame).

• Frame of Reference: A coordinate system combined with a time scale used to describe motion.

Relative Velocity in One Dimension

• When point $P$ is moving relative to reference frame $B$, and $B$ is moving relative to reference frame $A$, the x-velocity of $P$ relative to $A$ is given by: $v_{P/A-x} = v_{P/B-x} + v_{B/A-x}$

  • $v_{P/A-x}$: Velocity of $P$ relative to $A$ (along the x-axis).

  • $v_{P/B-x}$: Velocity of $P$ relative to $B$ (along the x-axis).

  • $v_{B/A-x}$: Velocity of $B$ relative to $A$ (along the x-axis).

Relative Velocity in Two or Three Dimensions

• The concept extends by using vector addition to combine velocities: $\vec{v}_{P/A} = \vec{v}_{P/B} + \vec{v}_{B/A}$

  • $\vec{v}_{P/A}$: Velocity vector of $P$ relative to $A$.

  • $\vec{v}_{P/B}$: Velocity vector of $P$ relative to $B$.

  • $\vec{v}_{B/A}$: Velocity vector of $B$ relative to $A$.

• This equation means that the velocity of an object $P$ seen by observer $A$ is the vector sum of $P$'s velocity relative to a moving frame $B$ and $B$'s velocity relative to $A$.

Examples of Relative Velocity

• EXAMPLE 3.13: Relative velocity on a straight road (1D)

  • Scenario: You drive north at constant $88 \text{ km/h}$. A truck approaches from the opposite lane (south) at $104 \text{ km/h}$.

  • Given: Your velocity relative to Earth ($v_{Y/E}$) $= +88 \text{ km/h}$. Truck's velocity relative to Earth ($v_{T/E}$) $= -104 \text{ km/h}$ (taking north as positive).

  • Tasks:

    • (a) Find the truck's velocity relative to you ($v_{T/Y} = v_{T/E} - v_{Y/E}$).

    • (b) Find your velocity relative to the truck ($v_{Y/T} = v_{Y/E} - v_{T/E} = -v_{T/Y}$).

    • (c) Analyze how relative velocities change after passing (they remain the same in magnitude but reverse direction for $v_{T/Y}$ and $v_{Y/T}$ due to the definition of relative velocities where $v_{A/B} = -v_{B/A}$).