Motion

Position of an object: where is it?

Velocity of an object: how fast and in what direction is it going?

Acceleration of an object: is the speed changing? Is the direction changing?

Motion described by vectors

Position of an object: r

Velocity of an object: v

Acceleration of an object: a

v a

This is for when Δt is very small, otherwise these are average quantities

Motion

Position of an object: r

Velocity of an object: v

Acceleration of an object: a

v a

All three vectors can be any direction and magnitude

The direction and magnitudes are defined in a coordinate system

A coordinate system is simply something we make up to be able to have reference points, e.g. maps.

Often, choosing a clever coordinate system makes problem solving easier

Motion in one dimension

This object’s velocity is not uniform…

So, there must be acceleration

When is velocity positive? Negative?

When is acceleration negative? Positive?

Does object change direction, or merely go faster and slower?

Look at a plot of some objects x-position vs. time:

Acceleration in one dimension

Acceleration may result in an object either speeding up or slowing down, but in either direction

If velocity of an object is constant, acceleration is 0

Acceleration in more than one dimension

speed of an object is changes, or its direction of motion, or both.

Acceleration

If the acceleration is constant, what does a plot of velocity vs. time look like?

•Is the velocity changing? Yes, constantly.

•When the rate of change of something is constant, the slope of a plot of that something is constant

•Plots with constant slopes are lines

Speed of an object (which starts at rest) subject to constant acceleration

speed grows faster at a constant rate

Acceleration

In the plot below, what’s the acceleration?

Δv = 8Δt = 2

In the plot, what is velocity as a function of time?

Equation of a line?

y = m x + b

v = 4 t

AccelerationIf the acceleration is constant, we can find the velocity as a function of time:

Acceleration

If the acceleration is constant, we can find the velocity as a function of time:

Definition of acceleration: a = Δv / Δt = (v-vo) / t (if to = 0)

What about the position? How does it change with time? Consider the definition of average velocity:

so

…which is the equation of a line with slope = a

Δx = x-x0 = v ΔtSo x = x0 + v Δt

and

Equations of motion (for Constant Acceleration)

Can use algebra to derive:

Gravity

Gravity is an acceleration, directed towards the earth. Near the surface of the earth, it’s

If the distance from the center of the earth doesn’t change much, it’s ~ constant.

Any object that’s falling near the ground has a constant acceleration (forgetting about air resistance) due to the Earth’s gravity.

Free Fall

The effects of air resistance are particularly obvious when dropping a small, heavy object such as a rock, as well as a larger light one such as a feather or a piece of paper.

However, if the same objects are dropped in a vacuum, they fall with the same acceleration.

Free Fall

So the equations of motion for free fall:

We just changed from x to y, and for a we put in -g.

The positive y direction has been chosen to be upwards. If it is chosen to be downwards, the sign of g would need to be reversed.

Example problem:

Man falls from roof, 5 meters high. How long before he hits the ground, and how fast is he going when he makes contact with ground?

-5 = -1/2 (9.8 ms-2) t2

v = (9.8)(1.01) = 9.9 m/s

5m