AP Physics Notes Newtonian Mechanics Unit

Representing Motion

Vector Diagrams: Depict object, forces acting on it, vector arrows (fancy term for “going thataway”

Free Body Diagrams: Object is a single point. Vector arrow TOUCH the point, and aim in their direction of action. (Gravity points down, etc.)

This is a vector diagram with a free body diagram superimposed on it.

Coordinate systems: Place the object’s motion into an x-y axis system. This is useful for finding direction and using geometry to solve for distance, velocity, or acceleration.

Geometry Rehash:

What if they ask you to find the angle (Ɵ)?

One-Dimensional Motion

Straight line, either up and down or side to side

Changes in direction are not curved (pivots)

Displacement

How far an object ends up from the starting point

NOT how far it travelled

When moving along the ground, represented by x, which is why you are no longer allowed to use x to represent any other unknown.

When moving up and down, represented by y, as in the y-axis

Equation for finding displacement:

=

“The change in position along∆ the푥 x-axis푥 푓is equal− 푥 to푖 the final position minus the starting position.” =

“The change in position along the y-axis is equal to the final position minus the starting position.” ∆푦 푦푓 − 푦푖

Velocity

Speed is how fast an object is moving

=

풅풊풔풕풂풏풄풆 풕풓풂풗풆풍풍풆풅 Speed can be constant—which푨풗풆풓풂품풆 means it does not풔풑풆풆풅 change 풕풊풎풆 풊풕 풕풐풐풌 Speed can be instantaneous—which means you know how fast something is going at the moment, but not its average speed

Speed is a scalar quantity, because you can measure it without knowing the direction of motion. However, it’s pretty useless.

Velocity is better than speed. It tells direction.

= =

풅풊풔풑풍풂풄풆풎풆풏풕 ∆풙 It is extremely simple to calculate풗풆풍풐풄풊풕풚 constant 풕풊풎풆velocity,풊풏풕풆풓풗풂풍 because∆풕 constant풂풏풅 풘풉풆풓풆 means풊풕 it풆풏풅풆풅 doesn’t풖풑 change. Velocity is called a vector, because it includes both magnitude (how fast) and direction. Vectors are represented with arrows. Physics tends to favor vectors. 

Graphing displacement vs time to get velocity

Graphing velocity versus time: to find the displacement.

TIME ALWAYS GOES ON THE X AXIS

Sne aky gra ph tric k: The area und er the line on the velocity vs. time graph is the displacement. Use ½ bh. Do you remember asking ‘when will I ever use this?’

Acceleration

Acceleration is any change in the velocity of an object.

Speeding up in the direction of motion is called positive acceleration.

Speeding up in the direction opposite of motion is called negative acceleration. (That’s an incredibly complicated way of saying ‘slowing down.’)

Acceleration clearly includes both magnitude and direction, so it’s also a vector.

The simple, nice, clean formula for acceleration, which you will be lucky if you ever use, is:

= = =

풄풉풂풏품풆 풊풏 풗풆풍풐풄풊풕풚 ∆풗 푣푓 − 푣푖 풂 Sneaky graph풄풉풂풏품풆 trick:풊풏 The풕풊풎풆 slope of∆ 풕the velocity푡푓 − 푡푖 versus time graph is the acceleration.

Picking apart the acceleration equation:

= 푣푓 − 푣푖 Hidden inside this equation풂 is the velocity, the displacement, ∆풕 and the time… = +

“The final velocity푣 푓is equal풗풊 to풂 ∆the풕 starting velocity plus the amount of time spent accelerating at a certain rate.”

Picking the velocity equation apart a little more:

= so = 푥푓−푥푖 푣푓−푣푖 ∆풕 To solve for xf: 풂 ∆풕 풂 ∆풕 = + + / ( ) ퟐ Say what??? Remember,푥푓 풙풊 풗 the풊∆풕 areaퟏ ퟐ풂under∆풕 the line in the velocity graph is the displacement. So

IF some jerk will give you velocity and an acceleration, but not a time…this equation relates displacement to accelration and velocity, without a required time unit.

Since you can say that

= , therefore = ( )/ …substitute your 푣푓−푣푖 equation for time and then simplify 풂 ∆풕 ∆풕 푣푓 − 푣푖 푎 = + + / ( ) ퟐ 푥푓 풙풊 풗풊∆풕 ퟏ ퟐ풂 ∆풕

= + 2 ퟐ

푣푓 풗풊 ퟐ풂∆풙