PHYS174 Exam 1: Kinematics and Dynamics Study Guide

Kinematics and Dynamics in Physics

Introduction

This study guide covers essential concepts in kinematics and dynamics, focusing on velocity vectors, projectile motion, tension forces, and Newton's laws. These topics are foundational for understanding motion and forces in introductory college physics.

Velocity Vectors and Relative Motion

Vector Representation of Velocity

Velocity is a vector quantity, meaning it has both magnitude and direction. When describing the motion of objects, it is important to represent their velocities accurately using vector diagrams.

• Velocity Vector: An arrow showing the direction and speed of an object's motion.

• Components: Velocity can be broken into horizontal (x) and vertical (y) components using trigonometry.

• Example: If a robot moves at 2 m/s north and another at 1.8 m/s 35° west of north, their velocity vectors can be drawn using the angle and magnitude.

Formula for velocity components:

• Horizontal component: $v_x = v \cos \theta$

• Vertical component: $v_y = v \sin \theta$

Relative Velocity

Relative velocity describes how fast one object is moving with respect to another. It is found by vector addition or subtraction.

• Relative Velocity: $\vec{v}_{A/B} = \vec{v}_A - \vec{v}_B$

• Application: To find the speed of one robot relative to another, subtract their velocity vectors.

• Example: If robot A moves north at 2 m/s and robot B moves 35° west of north at 1.8 m/s, use vector subtraction to find their relative speed.

Projectile Motion

Projectile motion occurs when an object moves in a curved path under the influence of gravity, after being launched.

• Key Equations:

  • Horizontal distance: $x = v_{0x} t$

  • Vertical distance: $y = v_{0y} t - \frac{1}{2} g t^2$

  • Maximum height occurs when vertical velocity is zero: $v_y = v_{0y} - g t = 0$

• Example: A robot drives off a ramp at an angle, and its speed as it leaves the ramp can be found using projectile motion equations and the time to reach the highest point.

Forces and Tension

Tension in Cables

Tension is the pulling force transmitted through a string, cable, or rope when it is pulled tight by forces acting from opposite ends.

• Maximum Tension: Occurs when the cable supports the maximum weight at a given angle.

• Formula: $T = \frac{W}{2 \cos \theta}$, where $W$ is the total weight and $\theta$ is the angle from the vertical.

• Example: For a gondola basket with a total weight of 5000 N and cable angle of 70°, calculate the tension in each cable.

Force Diagrams and Newton's Second Law

Newton's Second Law states that the net force on an object is equal to its mass times its acceleration.

• Equation: $\vec{F}_{\text{net}} = m \vec{a}$

• Application: When a cart is pulled by a force and a mechanical arm applies an additional force, draw all forces acting on the cart and use the equation to find acceleration.

• Example: A cart of mass 4000 kg is pulled with a tension $F_T$ and a force $F_P = 1800$ N is applied to accelerate it at $0.8$ m/s$^2$.

Resolving Forces at Angles

When forces are applied at angles, resolve them into components parallel and perpendicular to the direction of motion or tension.

• Component Formulas:

  • Parallel component: $F_{\parallel} = F \cos \theta$

  • Perpendicular component: $F_{\perp} = F \sin \theta$

• Example: If a tension force is applied at a 30° angle, calculate the effective force in the direction of motion.

Applications in Sports Physics

Ice Hockey Puck Motion

Analyzing the motion of a hockey puck involves concepts of kinematics, including displacement, velocity, and acceleration.

• Horizontal Displacement: $d = v t$

• Projectile Launch: When a puck is launched off the ice, its horizontal travel can be calculated using its initial speed and height.

• Banking Off Boards: The angle at which a puck strikes the boards can be found using trigonometry, given the distances involved.

• Acceleration: If a puck accelerates after bouncing, use $v_f = v_i + a t$ to find its final speed.

Summary Table: Key Equations and Concepts

Concept	Equation	Application
Velocity Components	$v_x = v \cos \theta$, $v_y = v \sin \theta$	Breaking velocity into x and y directions
Relative Velocity	$\vec{v}_{A/B} = \vec{v}_A - \vec{v}_B$	Speed of one object relative to another
Projectile Motion	$y = v_{0y} t - \frac{1}{2} g t^2$	Vertical displacement under gravity
Tension in Cable	$T = \frac{W}{2 \cos \theta}$	Maximum tension in supporting cables
Newton's Second Law	$\vec{F}_{\text{net}} = m \vec{a}$	Relating force, mass, and acceleration
Force Components	$F_{\parallel} = F \cos \theta$, $F_{\perp} = F \sin \theta$	Resolving forces at an angle

Best Practices for Solving Physics Problems

• Draw clear diagrams showing all forces and velocity vectors.

• Label all quantities, including units and directions.

• Break vectors into components before performing calculations.

• Apply relevant equations systematically, showing all steps.

• Check your final answer for correct units and reasonable magnitude.

Additional info: Some context and equations have been inferred to provide a complete and self-contained study guide for the exam topics.