Acceleration And Equations Of Motion 2

Magnitudes

Average speeds

Speed ms-1

Light 3x104

Electron around a nucleus

22

Earth around the Sun

1.5

Jet airliner 3x108

Typical car speed 1x10-3

Sprinter 2.5x102

Walking speed 2.2x106

Snail 10

Speed ms-1

Light 3x108

Electron around a nucleus

2.2x106

Earth around the Sun

3x104

Jet airliner 2.5x102

Typical car speed 22

Sprinter 10

Walking speed 1.5

Snail 1x10-3

Acceleration

• When an object's velocity changes, it accelerates.

• Acceleration shows the change in change in velocityvelocity during a period of time.

• Acceleration = change in velocity / time

• ms-2 ms-1 s

“change in velocity”

∆ - this symbol is a Greek (capital) letter, called delta. We use it in physics (and maths) to mean “the change in” a quantity.

∆ - we use it to save us from writing “change in” all the time!

∆ - To calculate the “change in” something we always minus the final quantity from the initial quantity

For example:

• Acceleration = change in velocity / time

..........can written much easier and quicker as:

a = ∆v / t

• A cyclist travels around a circular training track at a constant speed but is considered to be accelerating.

• Discuss!

QUESTION one: If a car accelerates from 5 m/s to 15 m/s in 2 seconds, what is the car's average acceleration?

QUESTION two: How long does it take Kitty to accelerate an object from rest to 10 m/s if the acceleration was 2 m/s2?

QUESTION three: Ella was running at 16m/s on the pavement. She starts to run on the sand and now has a speed of 10m/s. This change in speed took 2 seconds. What is her “acceleration”?

Warm up questions

QUESTION one: If a car accelerates from 5 m/s to 15 m/s in 2 seconds, what is the car's average

acceleration?

Initial velocity = 5 m/sFinal velocity = 15 m/sTime = 2 sa = ? m/s/s

a = ∆v / t = (15 – 5) / 2 = 10 / 2 = 5 m/s/s

How long does it take Kitty to accelerate an object from rest to 10 m/s if the acceleration was 2 m/s2?

Initial velocity = 0 m/sFinal velocity = 10 m/sAcceleration = 2 m/s2

Time = ?s

a = ∆v / t so t = ∆v / a = (10 – 0) / 2 = 10 / 2 = 5s

Clio was running at 16m/s on the pavement. She starts to run on the sand and now has a speed of 10m/s. This change in speed took 2

seconds. What is her “acceleration”?

Initial velocity = 16 m/s Final velocity = 10 m/sTime = 2 s“Acceleration” = ? m/s/s

a = ∆v / t = (10 – 16) / 2= - 6 / 2

= - 3 m/s/sThe minus tells us that it is deceleration!!

Slightly more complex..1. Harriet is riding her bike at 4m/s when she

decides to go a bit faster. She accelerates at 3m/s/s for 5s. What is her final velocity?

2. Tizzy is walking home at 2m/s when she sees the burglar coming out of a house, she gives chase and accelerates at a rate of 2ms-2. If it takes her 4s to accelerate what is her final speed?

3. The burglar has eaten too many pies and cannot accelerate as fast. He goes from stationary to 1m/s in 3s. What is his acceleration?

4. Will Tizzy catch the burglar?

Hattie is riding her bike at 4m/s when she decides to go a bit faster. She accelerates at 3m/s/s for 5s. What is

her final velocity?

Acceleration = 3 m/s/sTime = 5sInitial velocity = 4m/sFinal velocity = ? m/s

a = ∆v / t∆v = a x t = 3 x 5 = 15m/s∆v = final velocity – initial velocity Final velocity = ∆v + initial velocity = 15 m/s + 4= 15 + 4= 19m/s

Tizzy is walking home at 2m/s when she sees the burglar coming out of a house, she gives chase and accelerates at a rate of 2ms-2. If it

takes her 4s to accelerate what is her final speed?

Acceleration = 2 ms-2

Time = 4sInitial velocity = 2m/sFinal velocity = ? m/s

a = ∆v / t∆v = a x t = 2 x 4 = 8ms-1

∆v = final velocity – initial velocity Final velocity = ∆v + initial velocity = 8ms-1 + 2ms-1

= 10ms-1

The burglar has eaten too many pies and cannot accelerate as fast. He goes from stationary to 1m/s in 3s. What is his acceleration?

Acceleration = ? ms-2

Time = 3s

Initial velocity = 0m/s

Final velocity = 1ms-1

a = ∆v / t

a = (1 – 0) / 3

a = 0.3ms-2

Acceleration due to gravity

When objects fall freely through the air on Earth do they fall at

(a)Constant speed

(b)Accelerate at a constant rate

(c) Accelerate at an increasing rate

(d)Decelerate

Acceleration due to gravity

When objects fall freely through the air on Earth they fall at:

(a)Constant speed

(b)Accelerate at a constant rate

At first an object will accelerate at a constant rate and then gradually – as drag builds up, it will fall at a constant speed.

Hand in your homework on GATSOs please!

• Put your name on it!

• TIME TO HAND OUT BOOKS!

Task

• In groups of 3 use A3 paper to draw a diagram of a sky diver jumping out of a plane and falling.

• Include all stages of the fall until he lands – his parachute works.

• Include all force arrows, forms of motion and whether forces are balanced or unbalanced:

You have 5minutes

stopwatch

• http://www.online-stopwatch.com/

Uniformly accelerated motion in a straight line

• The melon clip:

• http://clipbank/espresso/clipbank/servlet/asset?assetID=5062

• The not so alive parachutist clip:

• http://clipbank/espresso/clipbank/servlet/asset?assetID=4450

Task

• Complete question on parachutist

• Homework: the rest of this sheet is to be completed for homework: Due in on Tuesday

Uniformly accelerated motion in a straight line

• There are 4 equations known as the “kinematic equations” or “equations of motion” that will completely describe the motion of a particle if it is travelling in a straight line and with uniform motion:

• You need to know where they come from as well as being able to use them!

• The equations of motion are valid only when acceleration is constant and motion is constrained to a straight line.

• We are over simplifying reality with a model so that we can an idea of the actual value.

What are the symbols and variables involved?

• s Displacement (m)

• u Initial velocity (ms-1)

• v Final velocity (ms-1)

• a Acceleration (ms-2)

• t Time (s)

Deriving the equations (There are 4 of them)

• You know the first one already but now it is written with v (final velocity) as the subject of the equation.

• a = v – u / t now written as

• v = u + at EQUATION ONE

Equation TWO: What is the displacement of this object?

• Note: v = u + at so v – u = at

Time (s)

Velocity (ms-1)

The area below the line of this graph tells you the displacement of the object:

Area of triangle: (v – u) x t / 2

Area of rectangle: ut

Total area: rectangle + triangle

= ut + (v-u)t / 2

= ut + at x t / 2

= ut + ½ at2

V

U

t

Equation Two

s = ut + ½at2

Equations so far:

v = u + at

s = ut + ½at2

Equation Three: What is the displacement, s of this object?

Time (s)

Velocity (ms-1)

The area below the line of this graph tells you the displacement of the object:

Total area: rectangle + triangle

s = ut + (v-u) ½ t

MULTIPLE OUT THE BRACKETS!

V

U

t

ut + (v-u) ½ t

ut + ½ vt - ½ ut

ut - ½ ut = ½ ut

½ vt + ½ ut Factorise:

½ t (v + u)

Equation Three

s = ½ t (v + u)

Equations so far:

v = u + ats = ut + ½at2

s = ½ t (v + u)

Equation 4:

• All three equations so far have t in them - it would be useful to get rid of it.

• Starting with the first equation, v = u + at we can write:

t = (v-u)÷a

This can be substituted into

the third, s = ½ (u+v) t :

s = ½ (u+v) (v-u)÷a

s = ½ (u+v) (v-u)÷a

• Expanding the brackets, and multiplying out the ½ and a:

2as = (u+v)(v-u)

uv - u2 + v2 + uv

2as = v² - u²

• This rearranges to: v² = u² + 2as

4 Equations of Motion

v = u + ats = ut + ½at2

s = ½ t (v + u) v² = u² + 2as

Example:

A car is travelling at 15ms-1 when it breaks and takes 50m to stop. Calculate the deceleration of the car.

s = 50m

u = 15ms-1

v = 0ms-1

a = ?ms-2

t = ?s

Which equation should you use?

HINT:

It has to be one without t in it!!!

4 Equations of Motion

v = u + ats = ut + ½at2

s = ½ t (v + u) v² = u² + 2as

v² = u² + 2as

A car is travelling at 15ms-1 when it breaks and takes 50m to stop. Calculate the deceleration of the car.

s = 50m

u = 15ms-1

v = 0ms-1

a = ?ms-2

t = ?s

Re arrange first:

v² = u² + 2as

a = (v² - u² ) / 2s

a = (0 – 152) / (2 x 50)

a = - 225 / 100

a = -2.25ms-2

Your turn:

• You MUST write down your answers to all questions using the format provided.

• Complete q1 – 7 (you might recognise q2!)

• Finish for homework