Science Starter

Take out your Ch.2 Homework and look over the problems. If you have any questions about the homework, now is the time to ask

We will spend 15 minutes going over any homework problems

Kinematics

Kinematics is the branch of mechanics that describes the motion of objects without necessarily discussing what causes the motion.

We will learn to describe motion in two ways.» Using equations» Using graphs

Particle

A particle is an object that has mass but no volume and occupies a position described by one point in space.

Physicists love to turn all objects into particles, because it makes the math a lot easier.

Position

How do we represent a point in space?

a) One dimension b) Two dimensions c) Three dimensions

(x)(x,y)(x,y,z)

Distance (d or x)

The total length of the path traveled by a particle.

“How far have you walked?” is a typical distance question.

SI unit: meter (m)

Displacement (x)

The change in position of a particle. “How far are you from home?” is a

typical displacement question. Calculated by… x = xfinal – xinitial

SI unit: meter (m)

Delta ( )

is a Greek letter used to represent the words “change in”. x therefore means “change in x”. It is always calculated by final value minus initial value.

Concept Question

Question: If x is the displacement of a particle, and d is the distance the particle traveled during that displacement, which of the following is always a true statement? a) d = |x|b) d < |x|c) d > |x|d) d > |x|e) d < |x|

Practice Problem

A particle moves from x = 1.0 meter to x = -1.0 meter.a) What is the distance d traveled by the particle?b) What is the displacement of the particle?

2.0 m

2.0 m @ 180°

Average Speed

How fast a particle is moving.

vavg = d t where: vavg = rate (speed) d = distance t = elapsed time SI unit:

m/s

Average speed is always a positive number.

Average Velocity

How fast the displacement of a particle is changing.

vavg = ∆x ∆t where:

vavg = average velocity∆x = displacement∆t = change in time

SI unit:m/s

Average velocity is + or – depending on direction.

Practice Problem

How long will it take the sound of the starting gun to reach the ears of the sprinters if the starter is stationed at the finish line for a 100 m race? Assume that sound has a speed of about 340 m/s.Answer: 0.29 s

Practice Problem

You drive in a straight line at 10 m/s for 1.0 km, and then you drive in a straight line at 20 m/s for another 1.0 km. What is your average speed?Answer: 13.3 m/s (this is probably NOT what you expected!)Always use the formula for average velocity; don’t just take an “average” of the velocities!

Acceleration (a)

Any change in velocity is called acceleration.

The sign (+ or -) of acceleration indicates its direction.

Acceleration can be…» speeding up» slowing down» turning

Uniform (Constant) Acceleration

In this course, we will generally assume that acceleration is constant.

With this assumption we are free to use this equation:

a = ∆v ∆t

SI Unit: m/s2

Acceleration has a sign!

If the sign of the velocity and the sign of the acceleration is the same, the object speeds up.

If the sign of the velocity and the sign of the acceleration are different, the object slows down.

Practice Problem

A 747 airliner reaches its takeoff speed of 180 mph in 30 seconds. What is its average acceleration?

Practice Problem

A horse is running with an initial velocity of 11 m/s, and begins to accelerate at –1.81 m/s2. How long does it take the horse to stop?

Big 3 Kinematic Equations

∆x = xi + vit + 1/2at2

vf = vi + at

vf2 = vi

2 + 2a(xf – xi)or …

Big 3 Kinematic Equations

x = vit + 1/2at2

vf = vi + at

vf2 = vi

2 + 2ax

Practice Problem

A plane is flying in a northwest direction when it lands, touching the end of the runway with a speed of 130 m/s. If the runway is 1.0 km long, what must the acceleration of the plane be if it is to stop while leaving ¼ of the runway remaining as a safety margin?

Practice Problem

You are driving through town at 12.0 m/s when suddenly a ball rolls out in front of you. You apply the brakes and decelerate at 3.5 m/s2.a) How far do you travel before stopping?b) When you have traveled only half the stopping distance, what is your speed?c) How long does it take you to stop?d) Sketch approximate x-vs-t, v-vs-t, a-vs-t graphs for this situation.

Practice Problem

On a ride called the Detonator at Worlds of Fun in Kansas City, passengers accelerate straight downward from 0 to 20 m/s in 1.0 second. a) What is the average acceleration of the passengers on this ride?b) How fast would they be going if they accelerated for an additional second at this rate?c) Sketch approximate x-vs-t, v-vs-t and a-vs-t graphs for this ride.

Practice Problem

Describe the motion of this particle.It is moving in the +x direction at constant velocity. It is not accelerating.

t

v

Practice Problem

Describe the motion of this particle.It is stationary.

t

v

Practice Problem

Describe the motion of this particle.It starts from rest and accelerates in the +x direction. The acceleration is constant.

t

v

Practice Problem

What physical feature of the graph gives the acceleration?The slope, because v/t is rise over run!

t

vv

tA

Ba = v/t

Practice Problem

Determine the acceleration from the graph.Ans: 10 m/s2

Practice Problem

Determine the displacement of the object from 0 to 4 seconds.Ans: 0Describe the motion.

The object is initially moving in the negative direction at –20 m/s, slows gradually and momentarily is stopped at 2.0 seconds, and then accelerates in the + direction. At 4.0 seconds, it is back at the origin, and continues to accelerate in the + direction.

Position vs Time Graphs Particles moving with no

acceleration (constant velocity) have graphs of position vs time with one slope. The velocity is not changing since the slope is constant.

Position vs time graphs for particles moving with constant acceleration look parabolic. The instantaneous slope is changing. In this graph it is increasing, and the particle is speeding up.

Uniformly Accelerating Objects

You see the car move faster and faster. This is a form of acceleration.

The position vs time graph for the accelerating car reflects the bigger and bigger x values.

The velocity vs time graph reflects the increasing velocity.

Position vs Time Graphs

This object is moving in the positive direction and accelerating in the positive direction (speeding up).

This object is moving in the negative direction and accelerating in the negative direction (speeding up).

This object is moving in the negative direction and accelerating in the positive direction (slowing down).

Pick the constant velocity graph(s)…

A

t

x

C

t

v

B

t

xD

t

v

(This is not in the notes.)

Draw Graphs forStationary Particles

x

t

Positionvs

time

v

t

Velocityvs

time

a

t

Accelerationvs

time

Draw Graphs forConstant Non-zero Velocity

x

t

Positionvs

time

v

t

Velocityvs

time

a

t

Accelerationvs

time

Draw Graphs for ConstantNon-zero Acceleration

x

t

Positionvs

time

v

t

Velocityvs

time

a

t

Accelerationvs

time

Kinematic Equations

v = vo + at»Use this one when you aren’t

worried about x. x = xo + vot + ½ at2

»Use this one when you aren’t worried about v.

v2 = vo2 + 2a(∆x)

»Use this one when you aren’t worried about t.

Practice problems

Landing with a speed of 115 m/s and traveling due south, a jet comes to rest in 7.00 x 102 m. Assuming the jet slows with constant acceleration, find the magnitude and direction of its acceleration.

Practice problems

When you see a traffic light turn red you apply the brakes until you come to a stop. If your initial speed was 12 m/s, and you were headed due west, what was your average acceleration during braking?

Suppose the car in the previous problem comes to rest in 35 m. How much time does this take?

Practice problems

Starting from rest, a boat increases its speed to 4.30 m/s with constant acceleration» (a) What was the boat’s average

speed?» (b) If it takes the boat 5.00 s to reach

this speed, how far has it traveled?

Practice problems

A cheetah accelerates from rest to 25 m/s in 6.2 s. Assuming constant acceleration,» (a) how far has the cheetah run in this

time?» (b) How far has the cheetah run in 3.1

s?

Model problem

A kid slides down a hill on a toboggan (a HAT?) with an acceleration of 3.0 m/s2. If he starts from rest, how far has he traveled in» (a) 1.0 s?» (b) 2.0 s?» (c) 3.0 s?

Model Problem

Two car drive on a straight highway. At time t = 0, car A passes mile marker 0 traveling due north with a speed of 28.0 m/s. At the same time, car B is 2.0 km south of mile marker 0 traveling at 30.0 m/s due south. Car A is speeding up with an acceleration of magnitude 1.5 m/s2, and car B is slowing down with an acceleration of magnitude 2.0 m/s2. Write x-vs-t equation of motion for both cars.

Practice Problem

Describe the motion of this particle.It is stationary.

t

x

Practice Problem

Describe the motion of this particle.It is moving at constant velocity in the + x direction.

t

x

Practice Problem

What physical feature of the graph gives the constant velocity?The slope, because x/t is rise over run!

t

xx

tA

Bvave = x/t

Practice Problem

Determine the average velocity from the graph.Ans: 1/3 m/s

x (m)

Practice Problem

t

x

Does this graph represent motion at constant velocity?

No, since there is not one constant slope for this graph.

Instantaneous Velocity

The velocity at a single instant in time.

Determined by the slope of a tangent line to the curve at a single point on a position-time graph.

Model Problem

A yacht cruising at 2.0 m/s is shifted into neutral. After coasting 8.0 m, the engine is engaged again and the yacht resumes cruising at a reduced speed of 1.5 m/s. How long did it take the yacht to coast the 8.0 m?

Practice Problem

You drive in a straight line at 10 m/s for 1.0 hour, and then you drive in a straight line at 20 m/s for 1.0 hour. What is your average velocity?Answer: 15 m/s (this is probably what you expected!)