Motion in 1D

Forces

Remember that an imbalance in forces results in an acceleration

If all forces are balanced we get a constant velocity

Because physics seeks to describe every aspect of the physical world. We must start with the most basic case of motion: 1D.

Distance

The total length of the path traveled by an object is called distance.

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

The SI unit of distance is the meter (m).

Distance is always positive. It has no direction. It is an example of a scalar.

Displacement (Dx)

The change in the position of a particle is called displacement.

“How far are you from home?” is a typical displacement question.

The SI unit for displacement is the meter.

Displacement has a direction. In 1D motion, it can be positive or negative. It is an example of a vector.

Calculation of displacement:f ix x x

A

B

Distance vs Displacement

Draw the distance and the displacement of a particle moving from A to B

A

B

Distance vs Displacement

Notice the displacement does not follow the track

Let’s Do a Lab!

Purpose: Figure out a way to make a cart move with an average velocity of as close to 0.200 m/s as possible. Use only the equipment provided. Photogate must be in PULSE mode.

Report: Write your BRIEF lab report. The section I want you to focus on in this lab report is the procedure section. Check my web site for examples of good procedures, and for the lab report guidelines.

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 velocity?

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 velocity?

Qualitative Demonstrations

1) Demonstrate the motion of a particle that has an average speed and an average velocity that are both zero.

2) Demonstrate the motion of a particle that has an average speed and an average velocity that are both nonzero.

3) Demonstrate the motion of a particle that has an average speed that is nonzero and an average velocity that is zero.

4) Demonstrate the motion of a particle that has an average velocity that is nonzero and an average speed that is zero.

Quantitative Demonstration

You are a particle located at the origin. Demonstrate how you can move from x = 0 to x = 5.0 and back with an average speed of 0.5 m/s.

What the particle’s average velocity for the above demonstration?

Graphical Problem

Demonstrate the motion of this particle.

t

x

Graphical Problem

Demonstrate the motion of this particle.

t

x

Graphical Problem

What physical feature of the graph gives the constant velocity from A to B?

t

xDx

DtA

Bvave = Dx/Dt

Graphical Review Problem

Demonstrate the motion of these two particles.

t

x

Graphical Problem

Demonstrate the motion of these two particle.

t

v

Graphical Problem

t

x

What kind of motion does this graph represent?

Graphical Problem

Can you determine average velocity from the time at point A to the time at point B from this graph?

t

x ABDx

Dt

vave = Dx/Dt

Graphical Problem: Determine the average velocity of this particle between 1 and 4 seconds.

x (m)

Graphical Problem: Determine the average velocity from the graph.

x (m)

Graphical Problem: Determine the average velocity between 1 and 4 seconds.

Graphical Problem: Determine the average velocity between 1 and 4 seconds.

Instantaneous Velocity

The velocity at a single instant in time. If the velocity is uniform, or constant,

the instantaneous velocity is the same as the average velocity.

If the velocity is not constant, than the instantaneous velocity is not the same as the average velocity, and we must carefully distinguish between the two.

Instantaneous Velocity

Draw a tangent line to the curve at B. The slope of this line gives the instantaneous velocity at that specific time.

t

xB Dx

Dt

vins = Dx/Dt

Practice Problem: Determine the instantaneous velocity at 1.0 second.

Practice Problem: Determine the instantaneous velocity at 1.0 second.

Acceleration (a)

Any change in velocity over a period of time is called acceleration.

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

Acceleration can be… speeding up slowing down turning

Questions

If acceleration is zero, what does this mean about the motion of an object?

Is it possible for a racecar circling a track to have zero acceleration?

Questions

If acceleration is zero, what does this mean about the motion of an object? Ans: It is not changing.

Is it possible for a racecar circling a track to have zero acceleration? Ans: Nope; the direction of the

velocity is changing.

Uniform (Constant) Acceleration

In most high school physics courses, we will generally assume that acceleration is constant or uniform.

With this assumption we are free to use this equation:

The SI unit of acceleration is the m/s2.

va

t

Acceleration in 1-D Motionhas a sign!

The sign indicates direction. Acceleration is therefore a vector.

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.

Qualitative Demonstrations

1) Demonstrate the motion of a particle that has zero initial velocity and positive acceleration.

2) Demonstrate the motion of a particle that has zero initial velocity and negative acceleration.

3) Demonstrate the motion of a particle that has positive initial velocity and negative acceleration.

4) Demonstrate the motion of a particle that has negative initial velocity and positive acceleration.

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

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?

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?

Graphical Problem

Demonstrate the motion of this particle. Is it accelerating?

t (s)

v (m/s)

0.50

Graphical Problem

Demonstrate the motion of this particle. Is it accelerating?

t

v

Graphical Problem

What physical feature of the graph gives the acceleration?

t

vDv

DtA

Ba = Dv/Dt

Practice Problem: Determine the acceleration from the graph.

Practice Problem: Determine the acceleration from the graph.

Free Fall

• Free fall is a great example of 1D motion. It is just in the Y direction.

• Problems usually ask for a speed or time it takes for the object to fall

• Think about the free body diagram of an object that is falling.

Free Fall Acceleration

• The acceleration due to gravity or 9.8 m/s2

• The acceleration is negative.• The velocity is negative if it is falling

towards the ground due to the fact that it is pointing down

Question

• Why is the acceleration due to gravity negative?

Example 1

• A block is dropped from 500 m. Assuming that the block starts from rest.– What is the final velocity just before it touches

the ground?– What is the time that it takes to get to the

ground?

Example 2

• A coin is dropped from a building. You know that the coin takes 2 seconds to fall. Assuming that the coin starts from rest.– What is the height of the building?

Example 3

• A ball is thrown up in the air with an initial velocity of 3 m/s. – What is the time that it takes to reach the

tallest point?– What is the height that it goes to?

E-instruction Question

• An egg is dropped from the top of a stair case that is 10 meters off the ground. How long does it take the egg to drop assuming that it starts from rest?

Kinematic Equations for uniformly accelerating objects

212

2 20 2 ( )

o

o o

v v at

x x v t at

v v a x