Projectile Motion, Circular Motion, and Relative Velocity – Study Notes

Projectile Motion and Two-Dimensional Kinematics

2D Motion: Equations and Concepts

Two-dimensional motion involves analyzing the movement of objects in both the x (horizontal) and y (vertical) directions. The equations of motion for each direction are solved independently, assuming constant acceleration.

• Position and velocity equations (constant acceleration):

• For the x-direction (horizontal): $x = x_0 + v_{0x} t + \frac{1}{2} a_x t^2$ $v_x = v_{0x} + a_x t$

• For the y-direction (vertical): $y = y_0 + v_{0y} t + \frac{1}{2} a_y t^2$ $v_y = v_{0y} + a_y t$

• Projectiles: Objects moving under the influence of gravity alone (free fall in y, constant velocity in x).

• Projectile motion is a combination of constant velocity along x and free-fall acceleration along y.

Example: A ball thrown horizontally from a cliff follows a parabolic path, with its horizontal and vertical motions analyzed separately.

Horizontal and Vertical Motion Independence

• x & y motion are independent: The horizontal and vertical components of motion do not affect each other.

• Projectile motion is a combination of constant velocity along x and free-fall acceleration along y.

Example: If a red ball is dropped and a yellow ball is fired horizontally from the same height at the same time, both hit the ground simultaneously.

Projectile Motion Diagram and Equations

• At the top of the trajectory, the projectile has zero vertical velocity ($v_y = 0$), but the horizontal velocity ($v_x$) remains constant.

• Horizontal motion: $v_x = v_{0x} = v_0 \cos \theta$; $a_x = 0$

• Vertical motion: $v_y = v_{0y} - gt = v_0 \sin \theta - gt$; $a_y = -g$

Example: An airplane moving with constant horizontal velocity drops a package. The horizontal distance before dropping is found using:

• Vertical: $y = y_0 + v_{0y} t + \frac{1}{2} a_y t^2$

• Horizontal: $\Delta x = v_{x} t$

Circular Motion

Uniform Circular Motion

Uniform circular motion refers to motion in a circular path at constant speed. The direction of velocity changes continuously, resulting in acceleration even if speed is constant.

• Centripetal acceleration: Always directed toward the center of the circle.

• Magnitude: $a_c = \frac{v^2}{r}$

• Velocity and acceleration are always perpendicular.

Key Point: In uniform circular motion, the only acceleration is radial (centripetal); there is no tangential acceleration.

Acceleration for Uniform Circular Motion

• Derived by considering the change in velocity vector as the object moves along the circle.

• Instantaneous acceleration (radial): $\vec{a} = -\frac{v^2}{r} \hat{r}$

• "Angular velocity": $\omega = \frac{v}{r}$

Arc Length of a Circle

• Arc length ($s$): The distance along the circle subtended by an angle $\Delta \theta$.

• Full circumference: $C = 2\pi r$

• Arc length: $\Delta s = r \Delta \theta$

• Rate of change: $\frac{d\theta}{dt} = \frac{v}{r}$

Small Angle Approximation

• For small $\theta$ (in radians): $\sin \theta \approx \theta$, $\cos \theta \approx 1$

• Useful for simplifying trigonometric expressions in physics problems involving small angles.

Acceleration on a Curved Path

For motion along a curved path, acceleration has two components:

• Radial (normal) acceleration ($a_r$): Due to change in direction, points toward the center. $a_r = \frac{v^2}{r}$

• Tangential acceleration ($a_t$): Due to change in speed (magnitude of velocity). $a_t = \frac{dv}{dt}$

• Total acceleration: $a = \sqrt{a_r^2 + a_t^2}$

Changing Speed on a Curved Path

• When speed increases, $a_t$ is parallel to velocity; when speed decreases, $a_t$ is opposite to velocity.

• Total acceleration vector is the vector sum of $a_r$ and $a_t$.

Circular Motion Exercises

• Given: Ferris wheel with radius $r = 10.0$ m, speed $v = 3.00$ m/s, tangential acceleration $a_t = 0.500$ m/s$^2$.

• Find: Total acceleration and its direction at the highest point.

• Radial acceleration: $a_r = \frac{v^2}{r} = \frac{(3.00)^2}{10.0} = 0.900$ m/s$^2$

• Total acceleration: $a = \sqrt{a_r^2 + a_t^2} = \sqrt{(0.900)^2 + (0.500)^2} = 1.03$ m/s$^2$

• Direction: $\theta = \tan^{-1}\left(\frac{a_t}{a_r}\right) = \tan^{-1}\left(\frac{0.5}{0.9}\right) = 29.1^\circ$ to the left of the normal

Relative Velocity

Definition and Frame of Reference

The velocity of a moving body as seen by an observer is called the relative velocity. A frame of reference is a coordinate system plus a time scale.

• Applications: Motion of airplanes, mid-air refueling, launching rockets, speed detectors.

Finding Relative Velocity

• Relative velocity in space: $\vec{v}_{P/A} = \vec{v}_{P/B} + \vec{v}_{B/A}$

• Example: A woman walks across a train moving at 4 mph (train) while she walks at 3 mph (relative to train). Her velocity relative to the ground is found by vector addition.

Example Problem

• A slow train travels at 4 mph in a straight line. A woman walks from one side of the train to the other at 3 mph. What is her speed and direction relative to an observer at rest?

• Solution: Use Pythagoras' theorem for perpendicular velocities: $v = \sqrt{(4\,\text{mph})^2 + (3\,\text{mph})^2} = 5\,\text{mph}$ $\theta = \tan^{-1}\left(\frac{3}{4}\right) = 37^\circ$

Summary Table: Components of Acceleration in Circular Motion

Component	Symbol	Direction	Magnitude	Cause
Radial (Centripetal)	$a_r$	Toward center	$\frac{v^2}{r}$	Change in direction
Tangential	$a_t$	Tangent to path	$\frac{dv}{dt}$	Change in speed
Total	$a$	Vector sum	$\sqrt{a_r^2 + a_t^2}$	Both effects

Additional info: These notes are based on slides from a university-level introductory physics course (Mechanics), covering projectile motion, circular motion, and relative velocity, with both conceptual and quantitative examples.