Kinematics in One Dimension

We will focus today on problem-solving.

Note: some problems are hard, some are not so hard. Part of the learning is recognizing quickly which problems are tough and which are not.

Remember that you can solve all of them!

Average Velocity vs. Average SpeedSpeed: how far an object travels in a given time interval

Velocity includes directional information:

(2-1)

Average speed does not take into account the direction of motion!

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

These graphs show (a) constant velocity and (b) varying velocity.

(2-3)

AccelerationAcceleration is the rate of change of velocity.

AccelerationAcceleration, like velocity, is a vector, although in one-dimensional motion we only need the sign.

The previous image shows positive acceleration; here is negative acceleration:

AccelerationThere 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.

Acceleration

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

UFO data!

The average velocity of an object during a time interval t is

The acceleration, assumed constant, is

Motion at Constant Acceleration Lab

Motion at Constant Acceleration

Start from what we know about the most general kind of problem – ones with constant acceleration.

We have all the equations we need to solve constant-acceleration problems.

Solving Problems

1.Read the whole problem and make sure you understand it.

2.Draw a diagram. Write down what you know.

3. Write down the unknowns that you need to find.

4. Plan an approach to a solution. (Pick an equation)

5.Does your answer make sense???

Free Fall Lab

The acceleration due to gravity at the Earth’s surface is approximately 9.80 m/s2. It is constant at any given spot!

Consider falling motion

Motion is straight up and/or straight down with respect to the earth.

Understanding the Problem…Understanding the Problem…

Start with the basics!Start with the basics!

When throwing a ball straight up,

which of the following is true

about its velocity v and its

acceleration a at the highest point

of its path?

A) both v = 0 and a = 0

B) v = 0, a = 0

C) v = 0, a = 0

D) both v = 0 and a = 0

Start with the basics…Start with the basics…

When throwing a ball straight up,

which of the following is true

about its velocity v and its

acceleration a at the highest point

of its path?

A) both v = 0 and a = 0

B) v = 0, a = 0

C) v = 0, a = 0

D) both v = 0 and a = 0

At the top the velocity must be zero by definition of “at the top” of the trajectory, but the acceleration due to gravity is constant along the entire trajectory

y

You throw a ball upward with an

initial speed of 10 m/s. Assuming

that there is no air resistance,

what is its speed when it returns

to you?

A) more than 10 m/s

B) equal to 10 m/s

C) less than 10 m/s

D) zero

Continue…Continue…

The ball is slowing down on the way up due to

gravity. Eventually it stops. Then it accelerates

downward due to gravity (again). Since a = g on

the way up and on the way down, the ball reaches

the same speed when it gets back to you as it had

when it left.

You throw a ball upward with an

initial speed of 10 m/s. Assuming

that there is no air resistance,

what is its speed when it returns

to you?

A) more than 10 m/s

B) equal to 10 m/s

C) less than 10 m/s

D) zero

Continue…Continue…

A little more depth…A little more depth…

Alice and Bill are at the top of a Alice and Bill are at the top of a

cliff of height cliff of height HH. Both throw a . Both throw a

ball with initial speed ball with initial speed vv00, Alice , Alice

straight straight downdown and Bill straight and Bill straight

upup. The speeds of the balls when . The speeds of the balls when

they hit the ground are they hit the ground are vvAA and and vvBB. .

If there is no air resistance, If there is no air resistance,

which is true?which is true?

A) vvAA < v < vBB

B) vvAA = v = vBB

C) vvAA > v > vBB

D) can’t tell

A little more depth…A little more depth…

Alice and Bill are at the top of a Alice and Bill are at the top of a

cliff of height cliff of height HH. Both throw a . Both throw a

ball with initial speed ball with initial speed vv00, Alice , Alice

straight straight downdown and Bill straight and Bill straight

upup. The speeds of the balls when . The speeds of the balls when

they hit the ground are they hit the ground are vvAA and and vvBB. .

If there is no air resistance, If there is no air resistance,

which is true?which is true?

A) vvAA < v < vBB

B) vvAA = v = vBB

C) vvAA > v > vBB

D) can’t tell

BillBillAliceAlice

HHvv00

vv00

vvAA vvBB

It often helps to add a picture…

A little more depth…A little more depth…

Alice and Bill are at the top of a Alice and Bill are at the top of a

cliff of height cliff of height HH. Both throw a . Both throw a

ball with initial speed ball with initial speed vv00, Alice , Alice

straight straight downdown and Bill straight and Bill straight

upup. The speeds of the balls when . The speeds of the balls when

they hit the ground are they hit the ground are vvAA and and vvBB. .

If there is no air resistance, If there is no air resistance,

which is true?which is true?

A) vvAA < v < vBB

B) vvAA = v = vBB

C) vvAA > v > vBB

D) can’t tell

BillBillAliceAlice

HHvv00

vv00

vvAA vvBB

Bill’s ball goes up but returns to the height of launch with the same speed v0 and the same acceleration g as Alice’s ball at launch. Therefore Bill’s ball lands with equal velocity to Alice’s.

Still more depth…Still more depth…

You drop a rock off a bridge.

When the rock has fallen 4 m,

you drop a second rock. As the

two rocks continue to fall, what

happens to their separation?

A) separation increases

B) separation is constant

C) separation decreases

D) impossible to tell

without more information

Still more depth…Still more depth…

You drop a rock off a bridge.

When the rock has fallen 4 m,

you drop a second rock. As the

two rocks continue to fall, what

happens to their separation?

A) separation increases

B) separation is constant

C) separation decreases

D) impossible to tell

without more information

At any given time, the first rock always has a greater velocity than the second rock, therefore it will always be increasing its lead as it falls. Thus, the separation will increase.

You try one…

Problem:Problem:In coming to a stop, a car leaves skid marks 92 m long on the highway. Assuming a deceleration of 7.00 m/s2, estimate the speed of the car just before braking.

Problem

A stone is thrown vertically upward with a speed of 12.0 m/s from the edge of a cliff 70.0 m high.

a)How much later does it reach the bottom of the cliff?

b)What is its speed just before hitting the ground?

c)What total distance did it travel?

Problem Solution

vv00

yy00

70 m

yy

+

gg

known information

unknowns

Challenge Problem

A flower pot is dropped, from rest, from the roof of a building. The windows of the building are of length h and spaced a distance h apart. If the pot takes a time t to pass from the top to the bottom of the top floor window, the time it will take to fall from the top to the bottom of the second window is about

a) ½ t

b) 2/3 t

c) ¼ t

d) 3/2 t

tough problem!Use the same approach: start with a picture with labeled known and unknown quantities, but note carefully that you need at least 4 equations to get to the relationship you want.

Summary• Kinematics is the description of how objects move with respect to a defined reference frame.

• Displacement is the change in position of an object.

• Average speed is the distance traveled divided by the time it took; average velocity is the displacement divided by the time.

• Instantaneous velocity is the limit as the time becomes infinitesimally short.

Summary

• Average acceleration is the change in velocity divided by the time.

• Instantaneous acceleration is the limit as the time interval becomes infinitesimally small.

• The equations for motion for constant acceleration are given in the text; there are four, each one of which requires a different set of quantities.

• Objects falling (or having been projected) near the surface of the Earth experience a gravitational acceleration of 9.80 m/s2.