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}).