1-D Kinematics

Objects can speed up, slow down, reverse directions.

All the equations of kinematics apply. (Equations all assume constant acceleration.)

2-D Kinematics

Objects can speed up, slow down, reverse directions, or turn and head at any angle.

All the same equations of kinematics apply.

Need to keep track of x-axis motion separately from y-axis motion. (Subscripts on equations.)

Most Important Point:

The motion in the x direction is completely independent from the motion in the y direction. No displacement, velocity, or acceleration along one axis has ANY effect on the displacement, velocity, or acceleration along the other axis.

Acceleration (continued)

2-D Speeding up, slowing down and changing direction

If the component of the acceleration vector that is parallel to the velocity vector points in the same direction as the velocity vector, the object is speeding up.

If the component of the acceleration vector that is parallel to the velocity vector points in the opposite direction of the velocity vector, the object is slowing down.

The component of the acceleration vector that is perpendicular to the velocity vector has NO effect on the speed of the object but does change the direction of travel. (It causes the object to turn.)

Projectiles

A special, though common, case of 2-D motion.

Some Vocab:

Projectile – The object that is shot, flung, kicked, thrown, hurled or otherwise projected into the gravitational field.

Trajectory – The path a projectile takes. (NOT the angle at which it’s projected though that does affect the trajectory.)

- Trajectory is NOT circular.

- Trajectory is parabolic.

Range – The horizontal distance the projectile travels.

Hang time – The amount of time the projectile is in the air.

We can discuss air resistance qualitatively but cannot include it in the computations.

- Air resistance is very complex. Depends on size,

shape and speed of projectile, density of air, etc…

- Air resistance will be ignored/considered negligible.
Projectiles (continued)

Separate into two parts: horizontal and vertical. And never the two shall meet EXCEPT FOR TIME.

x-direction (horizontal)

- constant velocity. v_0x = v_x = constant

- zero acceleration

- ONE equation: v_x = x / t

y-direction (vertical)

- constantly changing velocity.

- constant acceleration = 9.8 m/s² downward

- any/all kinematics equations may be needed

Velocities at any angle must be resolved into x and y components. (Most typically, the initial velocity.)

For horizontally projected projectiles (angle = 0^o),

v_initial = v_0x = v_x

v_0y = 0 m/s

Projectiles (continued)

Some helpful hints:

If a projectile takes off and lands at the same level (and in the absence of air resistance), the projectile’s trajectory is perfectly symmetric. Recognize:

- time going up = time coming down

- initial upward velocity magnitude (v_0y) =

final downward velocity magnitude (v_y)

- angle going up = angle coming down

- It might be possible and easier to just do half the

problem and then double the time and range.

For any projectile projected upward (at any angle),

v_y at the top = 0 m/s.

Don’t lose track of the signs. Usual convention is that up is positive and down is negative.

“g” IS A CONSTANT ACCELERATION VECTOR POINTING STRAIGHT DOWN AT ALL POINTS ALONG THE PROJECTILE’S PATH.

Projectiles (continued)

Some helpful hints continued:

If an upwardly directed projectile lands at a level below the level from which it takes off, you may want to use symmetry (see notes above and example in text) to divide the problem into a couple parts:

Part 1) A symmetric up and down (to same level) problem.

Part 2) A downwardly directed projectile problem.

Projectiles projected at complimentary angles have exactly the same range (if take off and landing is at same level.)

All things being equal (initial speed, take off and landing levels), an angle of 45^o will give the greatest range for a projectile.