Chapter 3
Kinematics in Two
Dimensions; Vectors
Units of Chapter 3
• Vectors and Scalars
• Addition of Vectors – Graphical Methods
• Subtraction of Vectors, and Multiplication of a
Vector by a Scalar
• Adding Vectors by Components
• Projectile Motion
• Solving Problems Involving Projectile Motion
• Projectile Motion Is Parabolic (qualitative)
• Relative Velocity
3-1 Vectors and Scalars
A vector has magnitude as
well as direction.
Some vector quantities:
displacement, velocity, force,
momentum
A scalar has only a magnitude.
Some scalar quantities: mass,
time, temperature
3-2 Addition of Vectors – Graphical Methods
For vectors in one
dimension, simple
addition and subtraction
are all that is needed.
You do need to be careful
about the signs, as the
figure indicates.
3-2 Addition of Vectors – Graphical Methods
If the motion is in two dimensions, the situation is
somewhat more complicated.
Here, the actual travel paths are at right angles to
one another; we can find the displacement by
using the Pythagorean Theorem.
3-2 Addition of Vectors – Graphical Methods
Adding the vectors in the opposite order gives the
same result:
3-2 Addition of Vectors – Graphical Methods
Even if the vectors are not at right
angles, they can be added graphically by
using the “tail-to-tip” method.
http://phet.colorado.edu/sims/vectoraddition/vector-addition_en.html
3-2 Addition of Vectors – Graphical Methods
The parallelogram method may also be used;
here again the vectors must be “tail-to-tip.”
3-3 Subtraction of Vectors, and
Multiplication of a Vector by a Scalar
In order to subtract vectors, we
define the negative of a vector, which
has the same magnitude but points
in the opposite direction.
Then we add the negative vector:
3-3 Subtraction of Vectors, and
Multiplication of a Vector by a Scalar
A vector V can be multiplied by a scalar c; the
result is a vector cV that has the same direction
but a magnitude cV. If c is negative, the resultant
vector points in the opposite direction.
22 sept: 3-4 Adding Vectors by
Components
Any vector can be expressed as the sum
of two other vectors, which are called its
components. Usually the other vectors are
chosen so that they are perpendicular to
each other.
3-4 Adding Vectors by Components
If the components are
perpendicular, they can be found
using trigonometric functions.
3-4 Adding Vectors by Components
The components are effectively one-dimensional,
so they can be added arithmetically:
3-4 Adding Vectors by Components
Adding vectors:
1. Draw a diagram; add the vectors graphically.
2. Choose x and y axes.
3. Resolve each vector into x and y components.
4. Calculate each component using sines and cosines.
5. Add the components in each direction to give
Resultant.
6. To find the length and direction of the resultant
vector, use:
Vector operations you need to
know
• How to resolve (“break”) a vector into x
and y components
– practice
• How to find resultant vector starting with x
and y components
– practice
• How to add vectors by adding x and y
components
– practice
Vector check for 28 sept
• Look at this velocity vector, V.
– What is Vx? Include units
– What is Vy? Include units
5 Oct: 3-5 Projectile Motion
A projectile
is an object
moving in
two
dimensions
under the
influence of
Earth's
gravity; its
path is a
parabola.
: 3-5 Projectile Motion
It can be understood by
analyzing the horizontal and
vertical motions separately.
3-5 Projectile Motion
The speed in the x-direction
is constant; in the ydirection the object moves
with constant acceleration g.
This photograph shows two balls
that start to fall at the same time.
The one on the right has an initial
speed in the x-direction. It can be
seen that vertical positions of the
two balls are identical at identical
times, while the horizontal
position of the yellow ball
increases linearly.
3-5 Projectile Motion
If an object is launched at an initial angle of θ0
with the horizontal, the analysis is similar except
that the initial velocity has a vertical component.
This is why we learned vectors!
3-6 Solving Problems Involving
Projectile Motion
Projectile motion is motion with constant
acceleration in two dimensions, where the
acceleration is g and is down.
Start with general kinematics and show these.
“The Big 10”
Problem solving steps
1. Draw and label the situation at two times; define in
each of x and y. Always draw the parabola.
2. Resolve any vector used into x and y components
3. Label knowns and unknowns (t,x,y,vx,vy,g)
–
–
You must choose the time interval for the problem
This time interval must have only freefall involved
4. Choose one dimension to start
5. Identify the correct kinematic equation for that x or y
Usually have to solve for t to link x and y
6.
7.
8.
9.
10.
Solve algebra for desired unknown
Find equation and solve for other dimension x or y
Check the units of the algebraic equations
Do the math with units
Check answer for sig figs
Practice, practice, practice
• The classics you will learn
– Cannon projectile -- on the level
– Car -- horizontal off a cliff
– Ball up off a cliff, find final velocity only
– Ball down off a cliff, find final velocity only
– Throw a ball up to a cliff ledge
3-7 Projectile Motion Is Parabolic (option)
In order to demonstrate that
projectile motion is parabolic,
we can use x= and y= equations
with substitution to see that:
This is
indeed the
equation for
a parabola.
The WWI German “Paris Gun”
The gun was capable of shooting a 94 kg (210 lb)
shell to a range of 130 km (81 mi) and a maximum
altitude of 40 km (25 miles, 131,000 ft) — the greatest
height reached by a human-made projectile.
At the start of its 170s trajectory, each shell from the
Paris Gun reached a speed of 1,600 m/s.
With these numbers, is the projectile behaving
according to our assumptions?
10 Oct: Classic problems chapter 3
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•
•
Cannon on the level
Car horizontal off a cliff
Ball up off a cliff, find final velocity only
Ball down off a cliff, find final velocity
Throw a ball up to a cliff ledge
12 oct: Labs in science classes
• Learning about science vs. Doing science
• Hypothesis: a well-defined proposition that
can be reproducibly tested
• Experiment: a reproducible process to test
a given hypothesis
• “lab” = experiment
• Prelab report: the preplanning for a lab
(see physics tools)
• Lab report: summary of the lab
(see physics tools)
3-8 Relative Velocity
We already considered relative speed in one
dimension; it is similar in two dimensions
except that we must add and subtract velocities
as vectors.
Each velocity is labeled first with the object, and
second with the reference frame in which it has
this velocity. Therefore, vWS is the velocity of the
water in the shore frame, vBS is the velocity of the
boat in the shore frame, and vBW is the velocity of
the boat in the water frame.
3-8 Relative Velocity
In this case, the relationship between the
three velocities is:
(3-6)
This is adding velocity
vectors just like
adding displacement
vectors to find overall
displacement
Summary of Chapter 3
• A quantity with magnitude and direction is a
vector.
• A quantity with magnitude but no direction is
a scalar.
• Vector addition can be done either graphically
or using components.
• The sum is called the resultant vector.
• Projectile motion is the motion of an object
near the Earth’s surface under the influence of
gravity.
Some questions for review
– Review symmetry of vdown and vup for vertical
motion
– What happens to tf or vf if h doubles?
– If throw up off a cliff and down off a cliff at
same vyo, why does object reach the ground
with same vy?
Quiz 4 october
• After reviewing ball off a cliff.
• I drop the ball with the GoMotion on the
ceiling. Falls in about 0.6s distance of
2.8m.
• Derive formula for time to fall
• Calculate numerical result for time
• Compare to observed time
Practice projectile problems
• HW
• Build classic list
Quiz on projectile lab
1. In the lab, there is one variable (condition
varied to see the effect of the change).
What is the variable, or what is varied?
2. What is the measured quantity in the lab;
what could change as the variable
changes?
3. Sketch the projectile path that you will
use in the lab.
Lab 2: projectile motion of a ball
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•
•
•
•
•
Use handout
Sketch the problem on the handout
Equipment and video analysis
Will you be here on Thursday for the full period?
Come to class Thursday ready to do the lab.
Groups:
– In work sessions.
Final quiz/problem
• Easier
– I throw the timer-ball at known angle, you
catch at time t. You know distance.
– What is vo?
– How high did the ball go?
• harder
– Set up in class, and solve for next class
– I throw this ball and it goes R distance in t
time.
– What is vo and q of the launch?
Quiz for Felix Buamgartner
jump -- 14 Oct 2012
1.
2.
3.
4.
Started at 128,100 ft above earth
Reached ~729 mph after ~48 seconds
Calculate average a.
Would you expect a to be constant?
Why?
5. Is it higher or lower than g? Why?
6. How far does he fall in 48 s?
=
6.79m/s2
Can’t know since a isn’t constant.
39.0km
= 326m/s