Fundamental Mechanics: Kinematics, Forces, and Friction (Physics 201 Exam #1 Study Guide)

Useful Information and Equations

Kinematic Equations

Kinematic equations describe the motion of objects under constant acceleration. They are essential for solving problems involving displacement, velocity, and acceleration.

• Displacement: $x_f = x_i + v_i riangle t + \frac{1}{2} a (\triangle t)^2$

• Final velocity: $v_f = v_i + a \triangle t$

• Velocity squared: $v_f^2 = v_i^2 + 2 a \triangle x$

Example: Calculating how far a car travels before stopping when brakes are applied.

Newton's Second Law

Newton's Second Law relates the net force acting on an object to its mass and acceleration.

• Equation: $F_{\text{net}} = \sum_i F_i = m a$

• Application: Used to analyze forces in free-body diagrams and solve for unknowns such as acceleration or force.

Weight

Weight is the force of gravity acting on an object.

• Equation: $w = m g$

• Where: $m$ is mass, $g$ is acceleration due to gravity ($9.8\,\text{m/s}^2$ on Earth).

Friction

Friction is a resistive force that opposes motion between two surfaces in contact.

• Maximum static friction: $f_{\text{max}} = \mu_s N$

• Kinetic friction: $f_k = \mu_k N$

• Normal force: $N$ is the perpendicular force exerted by a surface.

• Coefficients: $\mu_s$ (static), $\mu_k$ (kinetic) are dimensionless and depend on materials.

Drag Force

Drag is a resistive force experienced by objects moving through a fluid (air or liquid).

• Equation: $D = \frac{1}{2} C \rho A v^2$

• Where: $C$ is drag coefficient, $\rho$ is fluid density, $A$ is cross-sectional area, $v$ is velocity.

Exam Questions and Applications

1. Locomotive Braking and Kinetic Friction

This problem involves a locomotive decelerating due to friction after the brakes are applied. It tests understanding of kinematics and friction.

• Given: Mass = 720,000 kg, Initial speed = 120 km/h, Deceleration = 2.65 m/s2

• Key Points:

  • Use kinematic equations to find stopping distance.

  • Apply friction equations to find coefficient of kinetic friction.

• Answers:

  • a. 209 m (distance traveled before stopping)

  • b. 0.270 (coefficient of kinetic friction)

Example: Calculating stopping distance and friction for a train.

2. Projectile Motion: Arrow Shot Horizontally

This question examines horizontal projectile motion, focusing on time of flight and vertical velocity upon impact.

• Given: Height = 1.5 m, Initial horizontal speed = 12 m/s

• Key Points:

  • Time to hit ground: Use $y = y_0 + v_{y0} t + \frac{1}{2} g t^2$ with $v_{y0} = 0$.

  • Vertical speed at impact: $v_y = g t$.

• Answers:

  • a. 0.55 s (time to hit ground)

  • b. -5.4 m/s (vertical component of velocity at impact)

Example: Arrow shot horizontally from a bow.

3. Sled Pulled at an Angle: Forces and Friction

This problem involves analyzing forces on a sled being pulled at an angle, including friction and normal force.

• Given: Mass = 23 kg, Pull angle = 32°, Coefficient of kinetic friction = 0.35, Speed = 2.0 m/s

• Key Points:

  • Draw free-body diagram: Include weight, normal force, friction, and pull force components.

  • Calculate normal force using vertical force balance.

• Answer: 185 N (magnitude of normal force)

Example: Sled pulled across grass with friction.

4. Hockey Puck on Inclined Ramp: Friction and Acceleration

This question tests understanding of forces on an inclined plane, including static and kinetic friction, and calculation of final speed.

• Given: Mass = 200 g, Incline angle = 40°, Length = 1.8 m, $\mu_s = 0.40$, $\mu_k = 0.30$

• Key Points:

  • Draw free-body diagram: Include gravity, normal force, friction.

  • Calculate acceleration and final speed using kinematics and force analysis.

• Answer: 3.8 m/s (final speed at bottom of ramp)

Example: Hockey puck sliding down a ramp with friction.

Summary Table: Key Equations and Their Applications

Equation	Physical Principle	Application Example
$x_f = x_i + v_i \triangle t + \frac{1}{2} a (\triangle t)^2$	Kinematics (displacement under constant acceleration)	Locomotive stopping distance
$F_{\text{net}} = m a$	Newton's Second Law	Force analysis in free-body diagrams
$w = m g$	Weight (force of gravity)	Normal force calculation for sled
$f_k = \mu_k N$	Kinetic friction	Friction between sled and grass
$D = \frac{1}{2} C \rho A v^2$	Drag force	Air resistance (not directly tested in these questions)