General Physics Exam 1 Review: Kinematics, Dynamics, and Forces

Kinematics

Displacement, Velocity, and Acceleration

Kinematics is the study of motion without considering its causes. It involves analyzing displacement, velocity, and acceleration of objects.

• Displacement is the change in position of an object. It is a vector quantity and can be calculated as the difference between final and initial positions.

• Velocity is the rate of change of displacement with respect to time. It can be average or instantaneous.

• Acceleration is the rate of change of velocity with respect to time.

Key Equations:

• $v = v_0 + at$

• $d = v_0 t + \frac{1}{2} a t^2$

• $v^2 = v_0^2 + 2ad$

• $a = \frac{v - v_0}{t}$

• $v_{avg} = \frac{d}{t}$

Example:

• A particle starts at rest and accelerates at $a_1$ for 5.0s, then continues for 6.0m at a new acceleration $a_2$, ending with a velocity $v_2$. Calculate displacement, acceleration, time, and average velocity for each segment.

Graphical Analysis of Motion

Graphs of position, velocity, and acceleration versus time are useful for visualizing motion.

• Areas under velocity-time graphs represent displacement.

• Slopes of position-time graphs give velocity; slopes of velocity-time graphs give acceleration.

Example:

• Given a velocity-time graph, determine times at rest, direction changes, average and instantaneous acceleration, and produce an acceleration-time graph.

Projectile Motion

Two-Dimensional Motion

Projectile motion involves objects moving in two dimensions under the influence of gravity, typically with no air resistance.

• Horizontal motion has constant velocity.

• Vertical motion has constant acceleration due to gravity ($g = 9.81\,\text{m/s}^2$).

Key Equations:

• $x = v_{0x} t$

• $y = v_{0y} t - \frac{1}{2} g t^2$

• $v_{y} = v_{0y} - g t$

Example:

• A baseball is hit and lands with a velocity at a certain angle below the horizontal. Find the initial velocity and maximum height.

• An object is launched horizontally or at an angle; calculate initial velocity, horizontal distance, maximum height, and velocity at a given point.

Dynamics: Forces and Newton's Laws

Newton's Laws of Motion

Newton's laws describe the relationship between forces and motion.

• First Law: An object remains at rest or in uniform motion unless acted upon by a net external force ($\Sigma F = 0$).

• Second Law: The net force on an object is equal to its mass times its acceleration ($\Sigma F = ma$).

• Third Law: For every action, there is an equal and opposite reaction ($F_{12} = -F_{21}$).

Friction

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

• Static friction prevents motion up to a maximum value: $F_{s,max} = \mu_s N$

• Kinetic friction acts during motion: $F_k = \mu_k N$

• $\mu$ is the coefficient of friction, $N$ is the normal force.

Example:

• Calculate friction forces and acceleration for a crate on an incline, using given coefficients of static and kinetic friction.

• Find the friction force and coefficient for a skier's skis on a slope.

• Determine the coefficient of kinetic friction for a sanding block pushed up a wall.

Forces in Systems and Tension

Systems involving multiple objects may require analysis of tension forces, especially in pulleys or cables.

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

• For systems with acceleration, use Newton's Second Law for each object.

Example:

• Calculate the tension below and above a crate in a two-crate system being lifted with acceleration.

Applications of Kinematics and Dynamics

Motion with Changing Acceleration

Some problems involve objects moving with different accelerations in different segments.

• Use kinematic equations for each segment and match boundary conditions (position, velocity).

Relative Motion and Pursuit Problems

Relative motion problems involve two or more objects moving with respect to each other, often requiring calculation of catch-up times or distances.

• Set up equations for each object's position as a function of time and solve for when they meet.

Example:

• A motorcycle accelerates to catch up to a car moving at constant velocity; calculate the required acceleration.

• A police car pursues a truck after a delay; calculate time, acceleration, and velocities.

Vertical Motion and Free Fall

Vertical motion under gravity is a special case of kinematics.

• Maximum height is reached when vertical velocity is zero.

• Total time in air can be found by analyzing ascent and descent.

Example:

• Calculate maximum height, time to reach it, total time in air, displacement, and average velocity for a ball thrown upward.

• Analyze the motion of a rocket that accelerates upward, then coasts after fuel runs out.

Graphical Analysis: Parachutist Motion

Velocity, Displacement, and Acceleration Graphs

Graphical analysis is essential for understanding motion, especially when acceleration changes (e.g., parachute opening).

• Velocity-time graphs show changes in speed and direction.

• Displacement-time graphs can be derived from velocity-time graphs.

• Acceleration-time graphs show intervals of constant or changing acceleration.

Example:

• Given a velocity-time graph for a parachutist, sketch corresponding displacement-time and acceleration-time graphs.

Equations and Constants to Know

Summary Table of Key Equations

Topic	Equation
Kinematic Quantities	$a = \frac{v - v_0}{t}$ $v = \frac{d}{t}$ $\text{speed} = \frac{\text{distance}}{t}$
Newton's First Law	$\Sigma F = 0$
Newton's Second Law	$\Sigma F = ma$
Newton's Third Law	$F_{12} = -F_{21}$
Weight	$W = mg$
Frictional Force	$F_{friction} = \mu N$
Trigonometric Relationships	$\sin\theta = \frac{\text{opp}}{\text{hyp}}$ $\cos\theta = \frac{\text{adj}}{\text{hyp}}$ $\tan\theta = \frac{\text{opp}}{\text{adj}}$

Additional info:

• All problems are representative of introductory college-level physics, focusing on kinematics, dynamics, friction, tension, and projectile motion.

• Answers are provided for each problem, but students should practice solving using the equations above.

• Graphical analysis is emphasized for understanding motion in different scenarios.