Worked Examples

The WIKI contains a large number of example problems covering all the major topics of Newtonian mechanics. All of these example problems illustrate the S.I.M. problem solving approach. The examples are organized below by the relevant major branch of the Hierarchy of Models. By navigating to the principle page for each subgroup of problems, you can see the problems grouped by common interactions, procedures and topics.

Examples from Motion and Acceleration

• Accelerate, Decelerate — Determining the relationships between position, velocity and acceleration from a position vs. time plot.
• An Exercise in Continuity — An introduction to continuously piecing together kinematic solutions for time intervals with different accelerations.
• A Velocity for Words — Put the motion described by these graphs into words.
• Campus Tour — Basic problem to illustrate graphical representation of position and velocity.
• Dwarf Mistletoe — Perhaps this parasitic plant should be called "Dwarf Missiletoe".
• Fan-Powered Ice Boat — Graphing 1D motion with constant acceleration.
• Fountain — Modern fountains are an excellent example of projectile motion.
• Keys Please — Keys moving in 1D freefall with or without initial velocity.
• Lissajous Figures and the Bowditch Pendulum — Image generated by a Pendulum with two natural Frequencies.
• Overdriving Headlights — How long can you drive at constant velocity before you have to hit the brakes, assuming standard night detection distances?
• Space Station — find the approximate magnitude of the acceleration experienced by the space station as a result of the gravitational pull of the earth.
• Speed Trap — Police car with constant acceleration must catch speeder with constant velocity.
• Throwing a Baseball 1 (The Basics) — How far will the ball travel horizontally from the instant it leaves your hand until the instant it first contacts the ground?
• Throwing a Baseball 2 (Vectors) — How fast is the ball moving just before it impacts the ground?
• Training Flight — In this example we will calculate acceleration, time, speed, and distance assuming constant acceleration.
• Where Do We Meet? — Two people moving in one dimension with constant speed are destined to meet – but where?

Examples from Momentum and Force

• Apparently I've Lost Weight — Finding apparent weight using normal force.
• Atwood's Machine — The standard pulley problem as an example of systems.
• Basics of Static Friction — An introduction to determining the size of the static friction force.
• Cannonballs in a Boxcar — Examining the concept of Center of Mass and Conservation of Momentum in Different Ways.
• Chain Reaction — A series of elastic collisions.
• Finding Normal — Several examples showing how to find the normal force in common situations.
• Head-on Collision — Compare the forces on the occupants of two cars in a 1-D totally inelastic collision.
• I'm Inclined to Tilt the Coordinates — Basic inclined plane problems illustrating the advantage of tilted coordinates.
• Is That Normal? — Several examples illustrating how to find the normal force in not-so-common situations.
• Let it Rain — Analyzing a continuous momentum flux (falling water).
• Momentum Transport — Analyzing a continuous momentum flux (water from a hose).
• Off the Wall — Simple problem illustrating the definition of impulse and the utility of an initial-state final-state diagram.
• Out of Bounds — A typical perfectly inelastic collision in 2-D.
• Pushing a Box — A person pushes a box of mass 15 kg along a smooth floor by applying a perfectly horizontal force F.
• Pushing a Box Some More — A person pushes a box of mass 15 kg along a smooth floor by applying a force F at an angle of 30° below the horizontal.
• Pushing a Box with Friction — Assuming the coefficient of kinetic friction between the box and the ground is 0.45, what is the magnitude of F?
• Pushing a Box with Friction Some More — A person pushes a box of mass 15 kg along a floor by applying a force F at an angle of 30° below the horizontal. There is friction between the box and the floor
• Pushing Two Boxes — A person pushes a box of mass 15 kg along a smooth floor by applying a perfectly horizontal force F. In the process, the 15 kg box pushes against another box with a mass of 10 kg and causes it to move.
• Rope Bridge — The tension in ropes supporting an object can sometimes be much larger than the object's weight.
• Skydiving — Explore the force from air resistance acting on a skydiver at various stages of the dive.

Examples from Mechanical Energy and Work

• Bungee Jump — Bungee jumps involve elastic and gravitational potential energy.
• Bungee Jumping for the Brave — A more complicated version of the Bungee Jump problem.
• Diagrams and Mechanical Energy — Simple examples showing the utility of initial-state final-state diagrams and energy bar diagrams.
• Give it a Kick — A standard example from work and energy.
• Path Independence — An illustration of the path independence of gravitational work.
• Roller Coaster Diet? — A good roller coaster uses significant turning accelerations to produce large swings in the rider's apparent weight.
• Spring has Sprung — Energy and springs.
• When 7000 hp Just Isn't Enough — Explore the kinematics of constant power (as opposed to constant force).
• Why are you always so negative? — Explore the reason that kinetic friction usually produces negative work.

Examples from Angular Momentum and Torque

• A ball hits a bar and sticks to it. — find the angular velocity and the velocity of the final object's (bar + ball) center of mass
• Ballistic Pendulum Revisited — A version of the ballistic pendulum problem in which rotational effects are important.
• Lost in Space — The dangers of angular momentum in outer space.
• Moment of Inertia of a Block — Evaluate the integral to find the moment of inertia of a rectangular block.
• Moment of Inertia of a Solid Sphere — Integrate to find the moment of inertia of a solid sphere.
• Not-So-Simple Pendulum — Compares the simple pendulum model with a slightly more detailed one.
• Rolling Coin — A Coin rolling on its edge with a slight tilt will trace out a circle. What is its radius?
• Spinning Top — The rapidly spinning Symmetric Top exhibiting Precession under the force of gravity (near-earth) is a classic Physics problem.
• Twirling Skater — Changes in Angular Velocity when the Moment of Inertia is changed, but no torque applied.

Multi-Concept Problems for Mixed Review

• Accelerating Flywheel — Acceleration of a symmetric object about a fixed axis under constant torque (single-axis).
• A Walk on the Pond — How far will two children slide after a perfectly inelastic collision?
• Banking the Curve — Two examples of drawing free body diagrams for objects navigating a banked curve.
• Big Ben — Check Parliament's math by calculating the period of Big Ben's pendulum.
• Capture Cross-Section — Calculation of Effective Cross-Section of a planet with gravity (interaction)
• Close the Gate — Classic example of static friction on a moving surface.
• Cue the Right-Angle Bracket — Learn a valuable shortcut for dealing with a specific kind of elastic collision.
• Down the Alley — Determine the final speed of a ball that is initially sliding without rotation.
• Down the Ramp — Find the acceleration of a ball rolling without slipping on an inclined plane.
• Down the Well — A mass falling while attached to a massive pulley.
• Locking Your Bike — Determine how the weight of a bicycle plus rider is divided between the wheels in various circumstances.
• Mass Between Two Springs — A case of Simple Harmonic Motion.
• Mass on a Vertical Spring — Another case of Simple Harmonic Motion, this time with gravity (near-earth) thrown in.
• Pull Harder! — Examine the work needed to change the radius of rotation of an object rotating in a circle.
• Rotating a Space Ship — Changes in Angular Velocity when the Moment of Inertia is changed, but no torque applied.
• Sliding Yardstick — What happens to a yardstick (or meter stick) supported by two fingers as those fingers are slowly moved toward each other?
• The Ladder Problem — A standard statics problem.
• Watch Your Head — Consider the impulse and average force delivered to the head of a player performing a "header" in soccer.