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

Projectile Motion

2D Motion and Projectiles

Projectile motion is a classic example of two-dimensional motion in physics, where the motion along the x-axis and y-axis are independent of each other. The horizontal motion occurs at constant velocity, while the vertical motion is influenced by constant acceleration due to gravity.

• Key Equations for 2D Motion:

• Position equations: $x = x_0 + v_{x0} t$ $y = y_0 + v_{y0} t - \frac{1}{2} g t^2$

• Velocity equations: $v_x = v_{x0}$ $v_y = v_{y0} - g t$

• Projectile Motion: - Horizontal motion: Constant velocity ($a_x = 0$) - Vertical motion: Constant acceleration ($a_y = -g$) - The two motions are solved independently.

• Example: An airplane drops a package from 100 m above ground with a horizontal velocity of 40 m/s. The time to fall is $t = \sqrt{\frac{2y_0}{g}} = 4.5$ s, and the horizontal distance is $\Delta x = v_{x0} t = 180$ m.

Circular Motion

Uniform Circular Motion

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

• Centripetal (Radial) Acceleration: - Directed toward the center of the circle - Magnitude: $a_r = \frac{v^2}{r}$ - In uniform circular motion, the only acceleration is centripetal.

• Instantaneous Acceleration: - Always perpendicular to instantaneous velocity in uniform circular motion.

• Key Point: Acceleration exists even at constant speed due to changing direction.

• Example: Roller coaster vertical loops are examples of uniform circular motion.

Non-Uniform Circular Motion

When the speed of an object in circular motion changes, there is an additional component of acceleration called tangential acceleration.

• Radial (Centripetal) Acceleration: $a_r = \frac{v^2}{r}$

• Tangential Acceleration: $a_t = \frac{dv}{dt}$

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

• Direction: The total acceleration vector is the vector sum of radial and tangential components.

• Example: A Ferris wheel with radius 10.0 m, speed 3.00 m/s, and tangential acceleration 0.500 m/s2 has total acceleration $a = \sqrt{(0.900)^2 + (0.500)^2} = 1.03$ m/s2.

Arc Length and Angular Velocity

The arc length of a circle subtended by an angle $\Delta \theta$ is $\Delta s = r \Delta \theta$. The angular velocity $\omega$ is defined as $\omega = \frac{d\theta}{dt}$.

• Circumference: $c = 2\pi r$

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

• Angular Velocity: $\omega = \frac{d\theta}{dt}$

Small Angle Approximation

For small angles ($\theta \approx 0$), the sine and cosine functions can be approximated:

• $\sin \theta \approx \theta$

• $\cos \theta \approx 1$

Acceleration on a Curved Path

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

• Radial (Normal) Component: - Due to change in direction - $a_r = \frac{v^2}{r}$ - Directed toward the center

• Tangential Component: - Due to change in speed - $a_t = \frac{dv}{dt}$ - Directed along the tangent

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

Relative Velocity

Definition and Applications

Relative velocity is the velocity of a moving body as seen by an observer, and depends on the observer's frame of reference.

• Frame of Reference: 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 is calculated using vector addition:

• $\vec{v}_{P/A} = \vec{v}_{P/B} + \vec{v}_{B/A}$

• Example: If a train moves at 4 mph and a passenger walks at 3 mph perpendicular to the train's motion, the speed relative to the ground is $\sqrt{4^2 + 3^2} = 5$ mph, and the direction is $\theta = \tan^{-1}(3/4) = 37^ ext{o}$.

Summary Table: Components of Acceleration in Circular Motion

Component	Direction	Formula	Cause
Radial (Centripetal)	Toward center	$a_r = \frac{v^2}{r}$	Change in direction
Tangential	Along tangent	$a_t = \frac{dv}{dt}$	Change in speed
Total	Vector sum	$a = \sqrt{a_r^2 + a_t^2}$	Both effects

Additional info:

• These notes cover topics from Chapter 3: Motion in Two or Three Dimensions, including projectile motion, circular motion, and relative velocity, which are foundational for understanding Newton's Laws and further topics in mechanics.