Kinematics in One Dimension

Topic 2

Lecture Outline

• Distance and Displacement• Speed and Velocity• Acceleration

Distance and Displacement• What is the different between distance

and displacement? • Displacement (blue line) is how far the

object is from its starting point, regardless of how it got there.

• Distance traveled (dashed line) is measured along the actual path.

• Distance – Scalar• Displacement – Vector

The displacement is written:

Movement to the left:

Displacement is positive.

Movement to the right:

Displacement is negative.

Speed and Velocity

• Speed - how far an object travel in a given time interval

• Average speed – the total distance traveled along its path divided by the time takes to travel this distance

• Velocity – a vector, signify both of the magnitude (how fast) and direction of an object

• Average velocity – the total displacement of an object divided by the time takes to travel to this point

Unit : meter/second (m/s)

• Time taken – 70 s

• In general

• Positive value – object moving along the +x axis• Negative value – object moving along the –x axis

• Direction is always the same as the displacement

Example 2-1: Runner’s average velocity.

The position of a runner as a function of time is plotted as moving along the x axis of a coordinate system. During a 3.00-s time interval, the runner’s position changes from x1 = 50.0 m to x2 = 30.5 m, as shown. What was the runner’s average velocity?

Example 2-2: Distance a cyclist travels.

How far can a cyclist travel in 2.5 h along a straight road if her average velocity is 18 km/h?

Instantaneous VelocityThe instantaneous velocity is the average velocity in the limit as the time interval becomes infinitesimally short.

Example 2-3: Given x as a function of t.

A jet engine moves along an experimental track (which we call the x axis) as shown. We will treat the engine as if it were a particle. Its position as a function of time is given by the equation x = At2 + B, where A = 2.10 m/s2 and B = 2.80 m.

(a)Determine the displacement of the engine during the time interval from t1 = 3.00 s to t2 = 5.00 s.

(b) Determine the average velocity during this time interval.

(c) Determine the magnitude of the instantaneous velocity at t = 5.00 s.

Acceleration

Acceleration is the rate of change of velocity.

In general

(Unit: m/s2)

Example: 5 m/s2 – velocity will increase 5 m/s in 1 second

Example 2-4: Average acceleration.

A car accelerates along a straight road from rest to 90 km/h in 5.0 s. What is the magnitude of its average acceleration?

Example 2-6: Car slowing down.

An automobile is moving to the right along a straight highway, which we choose to be the positive x axis. Then the driver puts on the brakes. If the initial velocity (when the driver hits the brakes) is v1 = 15.0 m/s, and it takes 5.0 s to slow down to v2 = 5.0 m/s, what was the car’s average acceleration?

There is a difference between negative acceleration and deceleration:

Negative acceleration is acceleration in the negative direction as defined by the coordinate system.

Deceleration occurs when the acceleration is opposite in direction to the velocity.

The instantaneous acceleration is the average acceleration in the limit as the time interval becomes infinitesimally short.

Instantaneous Acceleration

• Like velocity, acceleration is a rate. • The velocity is the rate at which the

displacement changes with time• The acceleration it the rate which the

velocity changes with time• Acceleration is “rate of rate”

Example 2-7: Acceleration given x(t).

A particle is moving in a straight line so that its position is given by the relation x = (2.10 m/s2)t2 + (2.80 m). Calculate

(a)its average acceleration during the time interval from t1 = 3.00 s to t2 = 5.00 s, and

(b) its instantaneous acceleration as a function of time.

You and your dog go for a walk to the You and your dog go for a walk to the

park. On the way, your dog takes many park. On the way, your dog takes many

side trips to chase squirrels or examine side trips to chase squirrels or examine

fire hydrants. When you arrive at the fire hydrants. When you arrive at the

park, do you and your dog have the same park, do you and your dog have the same

displacement?displacement?

1) yes

2) no

ConcepTest 2.1ConcepTest 2.1 Walking the DogWalking the Dog

You and your dog go for a walk to the You and your dog go for a walk to the

park. On the way, your dog takes many park. On the way, your dog takes many

side trips to chase squirrels or examine side trips to chase squirrels or examine

fire hydrants. When you arrive at the fire hydrants. When you arrive at the

park, do you and your dog have the same park, do you and your dog have the same

displacement?displacement?

1) yes

2) no

Yes, you have the same displacement. Since you and your dog had

the same initial position and the same final position, then you have (by

definition) the same displacement.

ConcepTest 2.1ConcepTest 2.1 Walking the DogWalking the Dog

Follow-up:Follow-up: Have you and your dog traveled the same distance? Have you and your dog traveled the same distance?

ConcepTest 2.2ConcepTest 2.2 DisplacementDisplacement

Does the displacement of an object Does the displacement of an object

depend on the specific location of depend on the specific location of

the origin of the coordinate system?the origin of the coordinate system?

1) yes

2) no

3) it depends on the

coordinate system

ConcepTest 2.2ConcepTest 2.2 DisplacementDisplacement

Since the displacement is the

differencedifference between two

coordinates, the origin does

not matter.

10 20 30 40 50

30 40 50 60 70

x 40 10 30

x 60 30 30

Does the displacement of an object Does the displacement of an object

depend on the specific location of depend on the specific location of

the origin of the coordinate system?the origin of the coordinate system?

1) yes

2) no

3) it depends on the

coordinate system

If the position of a car is If the position of a car is

zero, does its speed have zero, does its speed have

to be zero?to be zero?

1) yes

2) no

3) it depends on the

position

ConcepTest 2.3ConcepTest 2.3 Position and SpeedPosition and Speed

If the position of a car is If the position of a car is

zero, does its speed have zero, does its speed have

to be zero?to be zero?

1) yes

2) no

3) it depends on the

position

No, the speed does not depend on position; it depends on the change

of position. Since we know that the displacement does not depend on

the origin of the coordinate system, an object can easily start at x = –3

and be moving by the time it gets to x = 0.

ConcepTest 2.3ConcepTest 2.3 Position and SpeedPosition and Speed

You drive for 30 minutes at 30

mi/hr and then for another 30

minutes at 50 mi/hr. What is your

average speed for the whole trip?

1) more than 40 mi/hr

2) equal to 40 mi/hr

3) less than 40 mi/hr

ConcepTest 2.6a ConcepTest 2.6a Cruising Along ICruising Along I

You drive for 30 minutes at 30

mi/hr and then for another 30

minutes at 50 mi/hr. What is your

average speed for the whole trip?

1) more than 40 mi/hr

2) equal to 40 mi/hr

3) less than 40 mi/hr

It is 40 mi/hr in this case. Since the average speed is distance/time

and you spend the same amount of time at each speed, then your

average speed would indeed be 40 mi/hr.

ConcepTest 2.6a ConcepTest 2.6a Cruising Along ICruising Along I

You drive 4 miles at 30 mi/hr and

then another 4 miles at 50 mi/hr.

What is your average speed for

the whole 8-mile trip?

1) more than 40 mi/hr

2) equal to 40 mi/hr

3) less than 40 mi/hr

ConcepTest 2.6b ConcepTest 2.6b Cruising Along IICruising Along II

You drive 4 miles at 30 mi/hr and

then another 4 miles at 50 mi/hr.

What is your average speed for

the whole 8-mile trip?

1) more than 40 mi/hr

2) equal to 40 mi/hr

3) less than 40 mi/hr

It is not 40 mi/hr! Remember that the average speed is distance/time.

Since it takes longer to cover 4 miles at the slower speed, you are

actually moving at 30 mi/hr for a longer period of time! Therefore,

your average speed is closer to 30 mi/hr than it is to 50 mi/hr.

ConcepTest 2.6b ConcepTest 2.6b Cruising Along IICruising Along II

ConcepTest 2.7ConcepTest 2.7 Velocity in One DimensionVelocity in One Dimension

If the If the averageaverage velocity is non-zero over velocity is non-zero over

some time interval, does this mean that some time interval, does this mean that

the the instantaneousinstantaneous velocity is velocity is nevernever zero zero

during the same interval?during the same interval?

1) yes

2) no

3) it depends

ConcepTest 2.7ConcepTest 2.7 Velocity in One DimensionVelocity in One Dimension

No!!! For example, your average velocity for a trip home

might be 60 mph, but if you stopped for lunch on the way

home, there was an interval when your instantaneous

velocity was zero, in fact!

If the If the averageaverage velocity is non-zero over velocity is non-zero over

some time interval, does this mean that some time interval, does this mean that

the the instantaneousinstantaneous velocity is velocity is nevernever zero zero

during the same interval?during the same interval?

1) yes

2) no

3) it depends

ConcepTest 2.8aConcepTest 2.8a Acceleration IAcceleration I

If the velocity of a car is non-zero If the velocity of a car is non-zero

((v v 00), can the acceleration of ), can the acceleration of

the car be zero?the car be zero?

1) Yes

2) No

3) Depends on the

velocity

ConcepTest 2.8aConcepTest 2.8a Acceleration IAcceleration I

Sure it can! An object moving with constantconstant velocityvelocity

has a non-zero velocity, but it has zerozero accelerationacceleration

since the velocity is not changing.

If the velocity of a car is non-zero If the velocity of a car is non-zero

((v v 00), can the acceleration of ), can the acceleration of

the car be zero?the car be zero?

1) Yes

2) No

3) Depends on the

velocity

When throwing a ball straight up, When throwing a ball straight up,

which of the following is true about which of the following is true about

its velocity its velocity vv and its acceleration and its acceleration aa

at the highest point in its path?at the highest point in its path?

1) both 1) both v = 0v = 0 and and a = 0a = 0

2) 2) v v 0 0, but , but a = 0a = 0

3) 3) v = 0v = 0, but , but a a 0 0

4) both 4) both v v 00 and and a a 0 0

5) not really sure5) not really sure

ConcepTest 2.8bConcepTest 2.8b Acceleration IIAcceleration II

y

At the top, clearly v = 0 because the ball has

momentarily stopped. But the velocity of the

ball is changing, so its acceleration is definitely

not zero! Otherwise it would remain at rest!!

When throwing a ball straight up, When throwing a ball straight up,

which of the following is true about which of the following is true about

its velocity its velocity vv and its acceleration and its acceleration aa

at the highest point in its path?at the highest point in its path?

1) both 1) both v = 0v = 0 and and a = 0a = 0

2) 2) v v 0 0, but , but a = 0a = 0

3) 3) v = 0v = 0, but , but a a 0 0

4) both 4) both v v 00 and and a a 0 0

5) not really sure5) not really sure

ConcepTest 2.8bConcepTest 2.8b Acceleration IIAcceleration II

Follow-up:Follow-up: …and the value of …and the value of aa is…? is…?