Projectile motion equations derived from motion with constant acceleration; separating the x and y components.

Golf ball fired directly upward from moving cart and caught later by the cart.

Proof that PM is a parabola; calculation of maximum height for PM with qualitative explanation.

Calculation of horizontal displacement for PM and angle α that maximizes it.

Demonstration of maximum horizontal displacement within calculated uncertainty for 45°.

Demonstration of the same displacement for 30° and 60° with different times.

Example of monkey falling from a tree when gun is fired at it; proof that monkey gets hit independent of velocity.

Proof that in the frame of reference of the monkey, the trajectory of bullet is straight, not an arc.

Demonstration of ball hitting monkey in flight.

Projectile equations of motion in x and y directions.

Worked example of projectile motion for KC135 jet; x, y, x', y' determined and graph analyzed.

Definition of a vector, displacement.

Addition, subtraction, negative of a vector; multiplying scalars and vectors.

Components of vectors in two or three dimensions, with examples; unit vectors; arbitrary vector in terms of unit vectors example.

Definition of dot product, with example; distributive law.

Definition of cross product of two vectors; right-hand rule; 2nd and 3rd order determinants; cross product examples.

Extension of one-dimensional motion to three dimensions. Includes average and instantaneous velocity, average and instantaneous acceleration, and constant acceleration.

Equations for ballistic motion in two dimensions. Includes examples.

List of steps for solving projectile motion problems in two-dimensions. Includes example.

Uniform circular motion; centripetal acceleration; non-uniform circular motion; reference frames; relativity of motion; problem-solving strategy for relative velocities with examples.

Overview of rectangular, spherical, and cylindrical coordinate systems.

Definition of a vector, displacement.

Addition, subtraction, negative of a vector; multiplying scalars and vectors.

Components of vectors in two or three dimensions, with examples; unit vectors; arbitrary vector in terms of unit vectors example.

Definition of dot product, with example; distributive law.

Definition for two vectors; right-hand rule; 2nd and 3rd order determinants; cross product examples.

Extension of one-dimensional motion to three dimensions. Includes average and instantaneous velocity, average and instantaneous acceleration, and constant acceleration.

Equations for ballistic motion in two dimensions. Includes examples.

List of steps for solving projectile motion problems in two-dimensions. Includes example.

Uniform circular motion; centripetal acceleration; non-uniform circular motion; reference frames; relativity of motion; problem-solving strategy for relative velocities with examples.

Performing vector addition using vector components; vector component notation.

Velocity and acceleration in multiple dimensions; special case of projectile motion; solving quadratic equations.

Projectile motion equations; circular motion equations; centripetal acceleration, with equation; dimensional analysis method.

Vector description of motion, including equations for position, velocity and acceleration; projectile motion; kinematic equations of motion in terms of x- and y- components; initial conditions; orbit equation with derivation.

Projectile motion experiment set-up and procedure.

3-part problem; maximizing distance for cannon to shoot when constrained by ceiling.

Solution (PDF)^#

Determining what conditions must apply for force vectors to add up to zero. Solution not included.

Velocity and acceleration of projectile motion. Solution not included.

Finding the landing point of a projectile on an inclined plane.

Solution (PDF)

Hitting a falling body with a projectile.

Solution (PDF)

Velocities in relatively inertial reference frames.

Solution (PDF)

Acceleration in relatively inertial reference frames.

Solution (PDF)

Speed and acceleration of a cart on a roller-coaster. Solution not included.

Projectile motion of two objects. Solution not included.

Magnitude and direction of velocity for a parabolic orbit. Solution not included.

Motion of a stone thrown at a falling person. Solution not included.

Speed and acceleration at the apex of projectile's path. Solution not included.

Motion of a fielder running to catch a softball.

Solution (PDF)

Motion of an object sliding off an inclined roof and falling to the ground.

Solution (PDF)

Finding maximum distance between two towns.

Solution (PDF)

Finding a parameter that makes two vectors orthogonal.

Solution (PDF)

Finding two unit vectors perpendicular to two given vectors.

Solution (PDF)

3-part 3D kinematics problem; finding v, |v|, a.

Solution (PDF)

3-part projectile problem; finding time and height of bullet hitting balloon, and time before firing (at fixed v, θ). Solutions are given at the end of each problem.

For boat crossing a flowing river, finding angle for direct crossing and for optimal time. Solutions are given at the end of each problem.

3-part 2D kinematics; finding point where walkers on square meet, distance to (0,0), and average velocity. Solutions are given at the end of each problem.

4-part problem; finding time and distance to impact, speed and location of impact. Solutions are given at the end of each problem.

Motion of a rock thrown upward from a bridge.

Solution (PDF)

Motion of a rock thrown from a bridge as seen from multiple reference frames.

Solution (PDF)

Finding and plotting the velocity and acceleration of a particle given equations for its position.

Solution (PDF)

Motion of a ball thrown from a moving boat.

Solution (PDF)

Motion of an acrobat catching a ball in mid-air.

Solution (PDF)

Comparing distance and hangtime for a punt at two different angles.

Solution (PDF)

Motion of an object sliding off an inclined roof and falling to the ground.

Solution (PDF)

Motion of a boat rowing across a river perpendicular to the current.

Solution (PDF)

Motion of a ball projected upwards and at an angle.

Solution (PDF)

Flight of a baseball.

Solution (PDF - 2.0 MB)

Motion of a thrown ball as seen by observers at different reference frames.

Solution (PDF)

Motion of an object that slides off the top step of a staircase.

Solution (PDF)

3-part projectile motion problem involving stones thrown into the air.

Solution (PDF)

3-part 3D kinematics and 2-part 1D kinematics problem about a moving particle.

Solution (PDF)

4-part projectile motion problem; finding highest point, final position, and speed.

Solution (PDF)