Kinematics and Relative Motion: Study Notes for Physics with Calculus

Motion and Kinematics

Introduction to Motion

Motion is a fundamental concept in physics, describing the change in position of objects over time. The study of motion is called mechanics, which is divided into kinematics (how objects move) and dynamics (why objects move).

• Kinematics focuses on describing motion using concepts such as position, velocity, and acceleration, without considering the causes of motion.

• Dynamics involves the forces and interactions that cause motion.

Frames of Reference and Coordinate Systems

To describe motion, it is essential to specify a frame of reference. A frame of reference is a coordinate system within which the position and motion of objects are measured. The same event can appear different in different frames of reference.

• Inertial frame: A non-accelerating frame where Newton's laws hold.

• Accelerated frame: A frame that is accelerating relative to an inertial frame.

Position and Displacement

The position of an object is described by a position vector \( \vec{r} \), which can be expressed in Cartesian coordinates as:

• \( \vec{r} = x \hat{i} + y \hat{j} \)

• In polar coordinates: \( x = r \cos \theta \), \( y = r \sin \theta \), \( r = \sqrt{x^2 + y^2} \), \( \theta = \tan^{-1}(y/x) \)

Displacement is the change in position of an object:

• \( \Delta \vec{r} = \vec{r}_2 - \vec{r}_1 = (x_2 - x_1) \hat{i} + (y_2 - y_1) \hat{j} \)

• The magnitude of displacement: \( \Delta r = \sqrt{(\Delta x)^2 + (\Delta y)^2} \)

• Distance is the total length of the path traveled, which may differ from displacement if the path is not straight.

Velocity and Speed

Velocity is the rate of change of displacement with respect to time. It is a vector quantity, having both magnitude and direction:

• Average velocity: \( \vec{v}_{avg} = \frac{\Delta \vec{r}}{\Delta t} = \frac{\Delta x}{\Delta t} \hat{i} + \frac{\Delta y}{\Delta t} \hat{j} \)

• Instantaneous velocity: \( \vec{v} = \lim_{\Delta t \to 0} \frac{\Delta \vec{r}}{\Delta t} = \frac{d\vec{r}}{dt} \)

• Magnitude: \( v = \sqrt{v_x^2 + v_y^2} \)

• Direction: \( \theta_v = \tan^{-1}(v_y / v_x) \)

Speed is the magnitude of velocity and is always positive:

• Average speed: \( v_{avg} = \frac{\text{distance}}{\Delta t} \)

• Instantaneous speed is the magnitude of instantaneous velocity.

Graphical Interpretation

• On a position-time (r-t) graph, the slope at any point gives the instantaneous velocity.

• On a velocity-time (v-t) graph, the area under the curve gives the displacement.

Acceleration

Acceleration is the rate of change of velocity with respect to time. It is a vector quantity:

• Average acceleration: \( \vec{a}_{avg} = \frac{\Delta \vec{v}}{\Delta t} = \frac{\vec{v}_2 - \vec{v}_1}{t_2 - t_1} \)

• Instantaneous acceleration: \( \vec{a} = \lim_{\Delta t \to 0} \frac{\Delta \vec{v}}{\Delta t} = \frac{d\vec{v}}{dt} \)

• On a velocity-time graph, the slope gives acceleration; on an acceleration-time graph, the area under the curve gives the change in velocity.

Deceleration refers to acceleration in the direction opposite to velocity, resulting in a decrease in speed.

Relative Motion and Galilean Relativity

Relative Velocity in One and Two Dimensions

Relative motion describes how the velocity of an object appears different depending on the observer's frame of reference. The Galilean transformation is used for adding velocities in classical mechanics:

• \( \vec{v}_{A/B} = \vec{v}_{A/C} + \vec{v}_{C/B} \)

• Example: If a bus moves at 5 m/s relative to the ground and a person walks at 1 m/s relative to the bus, the person's velocity relative to the ground is 6 m/s (forward) or 4 m/s (backward, if walking in the opposite direction).

Frame	Velocity (m/s)
Bus relative to Earth	5
Person relative to Bus	+1 (forward), -1 (backward)
Person relative to Earth	6 (forward), 4 (backward)

Relative Velocity: Car and Truck Example

When two vehicles move along the same straight road, their velocities relative to each other can be found using:

• \( \vec{v}_{T/C} = \vec{v}_{T/E} - \vec{v}_{C/E} \)

• Example: If a truck moves at 100 m/s and a car at 150 m/s (both relative to the ground), the velocity of the truck relative to the car is -50 m/s (i.e., the truck appears to move backward at 50 m/s from the car's perspective).

Relative Velocity in Two Dimensions: Boat and River Example

When motion occurs in two dimensions, such as a boat crossing a river with a current, vector addition is used:

• \( \vec{v}_{B/G} = \vec{v}_{B/W} + \vec{v}_{W/G} \)

• Where \( \vec{v}_{B/G} \) is the boat's velocity relative to the ground, \( \vec{v}_{B/W} \) is the boat's velocity relative to the water, and \( \vec{v}_{W/G} \) is the water's velocity relative to the ground.

• Example: If the boat moves north at 10 m/s relative to the water and the river flows east at 3 m/s, the boat's velocity relative to the ground is:

$\vec{v}_{B/G} = 10\,\hat{j} + 3\,\hat{i}$ $|\vec{v}_{B/G}| = \sqrt{10^2 + 3^2} = \sqrt{109} \approx 10.44\,\text{m/s}$ $\theta = \tan^{-1}\left(\frac{3}{10}\right) \approx 16.7^\circ$ east of north

Summary Table: Key Kinematic Quantities

Quantity	Symbol	Definition	Equation
Position	\( \vec{r} \)	Location in space	\( \vec{r} = x \hat{i} + y \hat{j} \)
Displacement	\( \Delta \vec{r} \)	Change in position	\( \Delta \vec{r} = \vec{r}_2 - \vec{r}_1 \)
Velocity	\( \vec{v} \)	Rate of change of displacement	\( \vec{v} = \frac{d\vec{r}}{dt} \)
Speed	\( v \)	Magnitude of velocity	\( v = |\vec{v}| \)
Acceleration	\( \vec{a} \)	Rate of change of velocity	\( \vec{a} = \frac{d\vec{v}}{dt} \)

Additional info:

• All equations are presented in SI units unless otherwise specified.

• Vector notation: \( \hat{i} \) and \( \hat{j} \) are unit vectors in the x and y directions, respectively.