Position, Velocity, Acceleration

Motion Notes

Where am I ? (Position)

Everything is located at a position in space X (m) (position in meters) To be consistent we need a frame of reference

(somewhere to place the origin) We can choose where to place the origin (it

doesn’t change the physics)

From Here to There (Distance vs. Displacement)

Distance- the length of the path that is taken (always a positive value) (odometer reading)

Displacement- the difference in positions (independent of the path taken)

Displacements can be positive or negative Totally independent of reference frame ΔX is the symbol for displacement ΔX = XF – X0

What time is it? (Time)

Object move through space and time The rate that an object moves through space

per unit time can be written in two ways Speed distance / time taken Velocity displacement / time taken Speed is a scalar quantity (magnitude only) while velocity

is a vector quantity (magnitude and direction)

Position vs. Time graphs (x vs. t)

Position (x) is always on the vertical axis, time (t) is on the horizontal axis

x (m)

t (s)0

Position vs. Time graphs (x vs. t) (cont.)

The slope represents the velocity on the object– The slope in measured in meters per second

The vertical intercept represents the starting position of the object (at t = 0 sec)

Position vs. Time graphs (x vs. t) (cont.)

When curved, the instantaneous velocity is the slope of a line tangent to the x vs. t curve

Mathematical model: X = V t + X0

Velocity vs. Time graphs (v vs. t)

Velocity (v) is on the vertical axis, time (t) is on the horizontal axis

v (m/s)

t (s)0

Velocity vs. Time graphs (v vs. t) (Cont.)

V vs. t graphs say NOTHING about where the object is

The area under the curve of a v vs. t graph is equal to the objects displacement

The slope of a v vs. t graph is the object acceleration

Acceleration

Acceleration is the change in velocity per unit time

Units (m/s) / s = m / s2

a = change in velocity / change in time

Acceleration (cont.)

Graphically this is the slope of a velocity vs. time graph

Graphical Views of Acceleration

Example #1, an object starts from rest at the origin and has a positive acceleration

Graphical Views of Acceleration (cont.)

Example #2, an object starts with a positive velocity from the origin and has a negative acceleration, eventually coming to a stop

Graphical Views of Acceleration (cont.)

Example #3, an object starts at a positive position with a negative velocity and has a negative acceleration

Graphical Views of Acceleration (cont.)

When the velocity and acceleration are in same direction, the object is “speeding up”

When the velocity and acceleration are in opposite directions, the object is “slowing down”

Mathematical Models

Mathematical models for Constant Acceleration

– V = V0 + a t– a = (VF - V0) / t– XF = ½ (V0 + VF) t + X0

– XF = X0 + V0 t + ½ a t 2

– VF 2 = V0 2 + 2 a (XF - X0)

Free Fallin’ (Free Fall motion)

When an object is dropped near the surface of the earth, it is said to be in free fall

Near the surface of earth, objects fall downwards with an acceleration of 9.8 m/s2 (assuming no air resistance)

Free fall is a special case of constant accelerated motion

Example #1

You jump horizontally off a cliff and hit the water 1.2 seconds later. How high was the cliff you jumped from?

Example #2

If you kick a ball into the air with an initial velocity of 15 m/s upwards, how much time does it spend in the air?