watch this movie to understand the difference between scalars and vectors. AS you watch the movie answers these questions. YOU NEED a graph paper + protractor. This worksheet is your next quiz. so pay attention.
here a small animation to understand the difference between distance and displacement.
A vector is a quantity that involves both
amplitude and direction
and obeys the communicative
law for addition. If A and B are vectors
we have A + B = B + A
see the figure below/
r is a vector. r could be a displacement if
you were to walk from the
tail of the vector to the tip. the length of r could
be proportional
(use a scale) to the distance you walked. r gives you also
a direction.
The displacement vector tells us "how far" and "which way"
(example: r = 50m@20 degrees Northeast)
Using a protractor and a ruler, define the vector r:
r = _____@_________
scale: 1cm = 10m
PART 2 : IN PHYSICS 1)READ: A vector r (see above) could be the velocity (in green below) of your car ("how fast", "which way")
Velocity is a vector. Velocity is one of the many quantities that
have direction, magnitude and unit..
Velocity tells you : "which way"and "how fast". Like: V = 30 mph@200
West of North
Speed is not a vector. speed is a scalar. The speedometer in the car gives you incomplete
information. It says nothing about which way
you are going. speed is a scalar. It has a magnitude and an unit.
2) r could be a force (in green below) . A force is a vector. For example you are dragging your little sister sitting on a sled. your pull makes an angle of 35 degrees with the horizontal:
3) Displacement is a vector. "Which way", "how far". D= 3 blocks @ North Distance is not a vector. Distance is a scalar. . It just tells you "how far" See the picture below. The distance covered is the distance along the red path but the displacement is the vector shown by the blue arrow connecting the starting point to the finish point. source: lessons on line: http://learn.uci.edu/oo/getOCWPage.php?course=OC0811004&lesson=006&topic=001&page=11
If you walk 1 block North turn back and walk 1 block south to the same starting points, your
total displacement is 1 block@North (+1 ) + 1 block@ South
(-1 ) = _______ but the distance covered is _________blocks.
4) What about these quantities? Which one are vectors? which ones are scalar ?
temperature ? energy ? acceleration ? force ? time ? volume ?
5) Highlight The vectors in pink and the scalar in yellow:
speed,
displacement, temperature, time,
acceleration, force, energy, velocity,
distance, volume, area
6) A car is moving from Start to Finish. Consider the image above (source: lessons on line, scienceS) 1 unit is 1 square. A) Find the distance covered from start to finish . distance = ________ units
B) trace the vector displacement . (hint: draw an arrow from start to finish)
C) Find the displacement vector , displacement = _________ units @ _________ degrees (hint: to find the magnitude, use Pythagorean theorem. To find the direction, use a protractor)
7) Imagine a bug crawling from A to B along the number line.
The displacement vector AB is a vector (an arrow) connecting A to B. A displacement is a change in position.
for all the examples: Find the displacement AB (a vector) i) AB = _____ units @ ________ or just - 6 (the negative sign shows the direction along the number line)
ii) AB = _____ units @ ________ or just + _____ iii) AB = _____ units @ ________ or just ___________
iv) AB = _____ units @ ________ or just ____________
8) the bug moves from A to D, but makes a stop along the way at B, C (moves from A to B to C to D ) Find the displacement AD
AD = _____ units @ ________ or just ____________
but the distance traveled is ___________ units
(hint: you can trace a vector (arrow) from A to D (initial to final) and count the magnitude of the vector and see its direction.
9) Find the displacement AD
AD = _____ units @ ________ or just ____________
but the distance traveled is _____________ units
10) same as 9) 11) Look at the bug below. ds represent the ________________ but the distance traveled is the length of the curly path.
With a ruler find the magnitude of ds = ____ cm (length of arrow) With a protractor find the direction of the vector ______ degrees (above x-axis) so the displacement ds = ____cm @ _______
12)
PART 3: ADDING PARALLEL VECTORS
Adding // vectors is easy. Remember
the number line? A positive sign indicates the right and
a negative sign indicates the left. Add the integers together.
1)You walk 3 blocks to the East (write A=+3) and 8 blocks
to the West (write B = -8).
What is you displacement ? (Find A + B).
2)You walk 3 miles to the North and 10 miles to the south what
is your displacement ?
3)You walk 10 blocks East and 10 West ?
4) You walk 20 blocks East and 10 West ?
5) you are in a train. the train is going at 100m/s@right and you walk at 1m/s@right.
a) what is your velocity (include direction) relative to a cow outside.
b) if you walk now at 1m/s @ left what is your velocity relative to the cow ?
5)
Practice adding vectors with the following applet. Work only in 1D (vectors going to the right/left or up/down). Might be a quiz in class.
6) A boat
can travel with a speed of 20.0m/s. If it going against a
current of 5.0m/s,
what is the velocity of the boat from a person standing
on the shore? Draw the vector representation of this problem
before solving.
Use integer: 20.0 m/s @right = + 20
5.0
m/s @ left = -5
7) A boat can
travel with a speed of 20.0m/s. If it going with a current
of 6.0m/s, what is the velocity of the boat from a person standing
on the shore? Draw the vector representation of this problem
before solving.
9)What is the angle between A and -A when they are drawn
from a common origin ?
10) What is the minimum number of vectors with equal magnitude whose
vector sum can be zero ?
11) Add these following vectors and DRAW THE VECTOR SUM . Example: 500N @ up + 800N @ down = 300N @ down Remember a vector needs a magnitude (300), a unit(N for newtons) and a direction (up or down).
12) Add this vectors. Example: 16 @ right + 20 @ left = 4 @ left = - 4 (negative sign means @ left) 4 is the magnitude and left is the direction
13) Fins the net force: A) 5N to the right and 5N to the left B) 4N to the right and 6N to the left C) 7N to the right and 5N to the left D) 6N to the right and 4N to the right
14) WRite if the following statements are true or false. A) Speed is velocity in a given direction B) The speed of a plane could be described as 300mi/h C) The velocity of a car can be described as 60 km/h to the north. D) Speed is a vector quantity E) velocity is a vector quantity
14) A) Is the following sentence true or false ? Five blocks south is an example of displacement B) Then complete: Five is the ___________, blocks is the __________ and, south is the ______________. C) what is the difference between displacement and distance ?
15)
What would you total displacement if you walked from your front door,
around the block, and then stopped when you reached your front door
again ?
16) What is the displacement of a cyclist who travels 1 mile north, then 1 mile east, and finally 1 mile south ?
17) What is the displacement of a cyclist that travels 3 miles south, 3 miles west and 3 miles north ?
PART4: HOW TO ADD VECTORS geometrically + algebraically (NOT DONE IF NO TIME) YOU NEED A RULER AND A PROTRACTOR do LAB pirates first. then lab trig
0) Do you remember our old friend Pythagorean theorem? here2)xus
1)You walk 3 miles North and then 7 miles east.
What is your displacement
from your starting point ? follow the steps: A) on a graph paper sketch the situation: First decide of a scale. like 1 cm = 1 mile. Then in a coordinate system, start from the origin and draw (use a ruler) an arrow = 7 cm @ right (depends on your scale) Then draw a second arrow (tail to head) on top of the previous one = 3cm@up. (that is attached to it) like you did for your lab with the pirates. B) To find the final displacement draw a third arrow (in a different color) from the origin to the head of the second vector. (the end of the second vector) . this third arrow is your final vector displacement.
C) Find the length (use a ruler) of this displacement vector and convert to miles.
D) Find the direction with your protractor. Find the angle between the final vector (the vector displacement) and the x-axis.
So displacement = D = ____ miles @___North of East
E) you could also use Algebra to find the same thing. I am going to show you how, step by step. Pythagorean theorem gives you the length of the vector. the length is the hypotenuse of the right angle triangle with legs 7 and 3. Check: length = hypotenuse = square root of (32 + 72) = _________ mile. Is it the same number ?
Now let's find the angle. You need your TI. The angle has to depend on the slope. If you increase the slope, the angle ________ and if you decrease the angle, the slope ________. There must be some king of relationship between the slope and the angle. And guess what ? there is !! Le'ts find the slope of the line that goes through the hypotenuse. (see image). ) Slope = rise/run = _____. This ratio in a right triangle has a name it is called. TAN (θ). tangent of θ. What we need now is to extract θ from the TAN. We need to undo TAN to extract θ. Here how to proceed: Get your TI. Go to MODE. move to DEGREE and ENTER. then enter : 2nd TAN 3/7) ENTER YOU ARE DOING TAN-1(θ) which extract θ from the ratio.
DO you get the same angle ? you should ! 2) refer to 1) A child is playing with a car on the floor of a train that is moving
eastward.
While the train travels 12.0 m (so 12m@east), the child pushes the car 2.6m northward
(2.6m@north) on the floor
of the train. A) sketch the situation. decide of a scale. 1cm = 1m. From the origin trace 1 arrow @ east and 1 arrow @ north. trace the vector sum by connecting the origin to the head of the second arrow. (different color please)
B)The use Pythagorean theorem to find the magnitude. Use TAN-1 (slope) to find the angle. D = ____m @ ____ North of East
3. A 110N force and a 55N force act on a point P. The 110N force acts due North.
THe 55 N force acts due east . WHat is the magnitude and direction of the resultant force (that is the net force that is the the sum of the 2 forces applied to point P. (You are just adding vectors) ?
A) sketch the situation. You start from the origin . 1 arrow @ east and 1arrow @ north. do it with a scale on a graph paper. maybe 50N = 5cm ? REmember order does not matter said the captain. trace the vector sum by connecting origin to the head of the north vector (by an arrow). B) The use Pythagorean theorem to find the magnitude. Use TAN-1 (slope) to find the angle.
Fnet = sum of the forces = ______ N @ _____ North of EAst
4.
a boat can travel 4.0 m/s in still water. It is in a river that flows
at 5.5 m/s southward.
If the boat heads eastward, directly across the river, what are the
direction and
magnitude of its total velocity ? (remember velocity is a vector. So total velocity is the vector sum of the 2 other velocities. Add the 2 vectors) A) sketch on scale the situation on a graph paper by adding the vector tail to head (see pirates lab) hint: 1 cm = 1m/s for example from origin 1 arrow east then 1 arrow down. head to tail. Connect the origin to the head of the second vector. B) Find the magnitude and direction don't forget to convert back to m/s (from cm) Final velocity = _____ m/s @ ________ South of East.
5. A plane is headed directly east at 340 mi/hr (mph) when the wind is from
the south at 45 mi/hr. What is its velocity (magnitude and direction) with respect to the ground ?
(that is find the vector sum = sum of the 2 velocities) hint: from the origin 1 vector 340mi/hr@east and then , attached to it, 45mi/ht@North. 50 mph = 1cm as a scale? Find the length of the final velocity. convert back to mph. Find the angle with the x-axis. Final velocity = _____ mph @ _____ North of east
6). 2 soccers players kick the ball at exactly the same time. One player's foot exerts a force of 66N north.
The other's foot exerts a force of 88N east. What is the magnitude and direction of the resultant force (net force = vector sum) on the ball ? hint: decide of a scale. Like 10N = 1cm. from the origin draw 88N@east as an arrow. then 66N@north attached to it. (arrow). Then trace the net force by connecting the origin to the end of the second vector.
Fnet = _________ N @ ______ North of East
7) A boat travels at 3.8 m/s@east and heads straight across a river 240m wide. The river flows at 1.6 m/s@down. Find the net velocity. You should know how to do by now ! Velocity net = _____ m/s @ _____South of EAst
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