Kinematics in two dimensions and vectors.pdf
AP* PHYSICS B DESCRIBING MOTION: KINEMATICS IN TWO DIMENSIONS &VECTORS The moment of truth has arrived! To discuss objects that move in something other than a straight line we need vectors. VECTORS Vectors have both magnitude AND direction while scalars have only magnitude. Velocity, displacement, force and momentum are vectors. Speed, mass, time and temperature are scalar quantities. ! Vectors--arrows drawn to show direction and the length of the arrow is % to magnitude ADDITION OF VECTORS Graphical, tedious method--tip to tail. If the motion or force is along a straight line, simply add the two or more lengths to get the resultant. More often, the motion or force is not simply linear. That’s where trig. comes in. You can use the tip to tail graphical method, BUT you’ll need a ruler and a protractor.
!
Resultant--the vector sum. D1 = 1.0 km East D2 = 5.0 km North DR = resultant and no, it’s not 6.0 km NE!! To get the resultant, draw a picture and use trigonometry to get BOTH the magnitude and direction of the resultant. REMEMBER to report the angle in the “lesser” fashion. 44E N of E rather than 46E E of N. DR = (D12 + D2 2 )½ = (100 + 25 km2)½ = 11.2 km Pythagorean’s theorem is your friend!
A P® is a registered trademark of the C ollege B oard. The C ollege B oard w as not involved in the production of and does not endorse this product. © 2009 by René M cC ormick. A ll rights reserved
Kinematics in Two Dimensions & Vectors
1
!
To get the direction (angle):
Use trig. functions-- “Oscar Had A Heap Of Apples” is my favorite mnemonic for sin, cos, and tan (Use your own favorite!) O H
=
Opposite = sin Hypotenuse
A = Adjacent = cos H Hypotenuse O A !
=
Opposite Adjacent
= tan
HINT: Use the data given, don’t be so cocky as to use the resultant you calculated, if you made an arithmetic error, you’re setting yourself up to make a second one! They gave me the opposite and adjacent sides to è, the angle I want to use in reporting my direction. Tan è =
Opposite Adjacent
= 5 (km / km leaves this unitless) = 0.5 10
è = tan -1 0.5 = 27E LOOK AT YOUR PICTURE! It’s 27E N of E [as opposed to 63E E of N–it is customary to report the “lesser” measure] ! !
All vectors have positive magnitude If a vector is NEGATIVE, the (!) refers to its direction
SUBTRACTION OF VECTORS !V means same magnitude, opposite direction
Kinematics in Two Dimensions & Vectors
2
MULTIPLICATION OF A VECTOR BY A SCALAR V × scalar quantity = magnified magnitude, same direction as before if scalar quantity is positive, opposite direction as before is scalar quantity is negative. ADDING VECTORS BY COMPONENTS We can “split” a vector into x-components and y-components and use a chart to solve for resultant vectors quite easily. What’s a component? Consider the vector V shown in diagram (a) below:
Start by drawing a set of axes at the “head” of the vector (its origin if you prefer). The x and y-components lie along their respective axes (b) and their magnitudes are equal to: Vy = V sin è Vx = V cos è You’ll come back to this method over and over between now and May! We’ll work some examples in a bit. For now, here’s more useful trig stuff with which to wow your geeky friends [or just me!] This Pythagorean Relationship also comes in handy from time to time: sin2 è + cos 2 è = 1
Law of sines:
A
c
B
b
a
C
Law of cosines: Both of these laws are handy when you have a triangle that is not a right triangle! Kinematics in Two Dimensions & Vectors
3
Example 1 A rural mail carrier leaves the post office and drives 22.0 km in a northerly direction to the next town. She then drives in a direction 60.0E south of east for 47.0 km to another town. What is her displacement from the post office?
Example 2 An airplane trip involves three legs, with two stopovers. The first leg is due east for 620 km; the second leg is southeast for 440 km; and the third leg is 53E south of west, for 550 km. What is the plane’s total displacement?
Kinematics in Two Dimensions & Vectors
4
PROJECTILE MOTION Tiger Woods hitting a golf ball, a thrown or batted baseball, kicked footballs, speeding bullets, and athletes doing the long jump or high jump are all examples of projectile motion. OBJECTS PROJECTED PARALLEL TO THE GROUND/SURFACE ! ! ! ! !
We will often ignore air resistance (be thankful!) We will only consider the motion after the projection (again be thankful!) Galileo was the first to analyze projectile motion. To keep it simple, we assume the motion begins at t = 0 and at the origin of an x,y coordinate system so xo = yo = 0 Treat the 2 components separately, and you can apply the kinematics equations from last chapter to each component
Let’s consider a tiny ball rolling off a table. It’s velocity vector at each instant is tangent to the path. !
Consider vy: Initially it’s zero, then gravity kicks in so now, it’s vy = !1/2 gt 2 (if you assign “up” as positive, then y + 8 and !9)
!
Consider vx: Initially it’s zero, it experiences no acceleration in the x-direction, therefore its velocity remains constant and vx = 0
!
Gravity prevails! An object projected horizontally will reach the ground in the same time as an object dropped vertically since the vertical motions are the same in each case. Dazzle your slacker friends with this new found knowledge!
OBJECTS PROJECTED AT AN UPWARD ANGLE Now you’ve gone and done it! You have introduced an initial vertical velocity component, vyo !
Because of the downward acceleration of gravity, vy continually decreases (slows) until the object reaches its highest point on its path, at which point vy = 0. Yep, you guessed it, now that sucker starts to accelerate downward
!
Good news! vx remains constant. Make peace with this concept. Now. Right now.
!
How can this be? (It know, it seems too easy and you are a suspicious lot by nature, huh?) The answer is: Simply because no horizontal force is acting on the projectile in the x-direction, velocity in the x-direction remains constant!
Kinematics in Two Dimensions & Vectors
5
SHALL WE JUMP OFF A CLIFF? !
Use the kinematic equations separately for each component, vertical and horizontal (see charts below)
!
Simplifications: ax = 0 ay = !g = ! 9.8 m/s2 Choose è relative to x-axis and: vxo = Vo cos è vyo = Vo sin è
Example 3 A movie stunt driver on a motorcycle speeds horizontally off a 50.0 m high cliff. How fast must the motorcycle leave the cliff-top if it is to land on level ground below, 90.0 m from the base of the cliff where the cameras are?
Kinematics in Two Dimensions & Vectors
6
Example 4 A football is kicked at an angle èo = 37.0E with a velocity of 20.0 m/s. Calculate a) the maximum height
b) the time of travel before the football hits the ground
c) how far away it hits the ground
d) the velocity vector at the maximum height
e) the acceleration vector at maximum height. Assume the ball leaves the foot at ground level.
Example 5 A child sits upright in a wagon which is moving to the right at constant speed as shown in the figure. The child extends her hand and throws an apple straight upward, while the wagon continues to travel forward at constant speed. If air resistance is neglected, where will the apple land?
Kinematics in Two Dimensions & Vectors
7
Example 6 A boy on a small hill aims his water-balloon slingshot horizontally, straight at a second boy hanging from a tree branch a distance d away. At the instant the water balloon is released, the second boy lets go and falls from the tree, hoping to avoid being hit. Show that he made the wrong move!
LEVEL RANGE FORMULA:
FOR USE WHEN y = yo ONLY! Where è is the angle made with the horizontal x-axis. You will always get 2 answers (roots) è and 90E ! è.
Example 8 Suppose the football in Ex. 4 was a punt and left the punter’s foot at a height of 1.00 m above the ground. How far did the football travel before hitting the ground? Set xo = 0 and yo = 0.
Kinematics in Two Dimensions & Vectors
8
PROJECTILE MOTION IS PARABOLIC That means y is a function of x an has the form y = ax ! bx2, where a & b are constants for any specific parabolic motion. Make friends with your graphing calculator... RELATIVE VELOCITY--The main reason we need vectors! Example 10 A boat’s speed in still water is 1.85 m/s. If the boat is to travel directly across a river whose current has speed 1.20 m/s, at what upstream angle must the boat head?
Example 11 The same boat now heads directly across the stream whose current is still 1.20 m/s. a) What is the velocity of the boat relative to the shore?
b) If the river is 110 m wide, how long will it take to cross and how far downstream will the boat be then?
Kinematics in Two Dimensions & Vectors
9
Example 12 A plane whose airspeed is 200 km/h heads due north. But a 100-km/h northeast wind (that is, coming from the northeast) suddenly begins to blow. What is the resulting velocity of the plane with respect to the ground?
Kinematics in Two Dimensions & Vectors
10
!
Resultant--the vector sum. D1 = 1.0 km East D2 = 5.0 km North DR = resultant and no, it’s not 6.0 km NE!! To get the resultant, draw a picture and use trigonometry to get BOTH the magnitude and direction of the resultant. REMEMBER to report the angle in the “lesser” fashion. 44E N of E rather than 46E E of N. DR = (D12 + D2 2 )½ = (100 + 25 km2)½ = 11.2 km Pythagorean’s theorem is your friend!
A P® is a registered trademark of the C ollege B oard. The C ollege B oard w as not involved in the production of and does not endorse this product. © 2009 by René M cC ormick. A ll rights reserved
Kinematics in Two Dimensions & Vectors
1
!
To get the direction (angle):
Use trig. functions-- “Oscar Had A Heap Of Apples” is my favorite mnemonic for sin, cos, and tan (Use your own favorite!) O H
=
Opposite = sin Hypotenuse
A = Adjacent = cos H Hypotenuse O A !
=
Opposite Adjacent
= tan
HINT: Use the data given, don’t be so cocky as to use the resultant you calculated, if you made an arithmetic error, you’re setting yourself up to make a second one! They gave me the opposite and adjacent sides to è, the angle I want to use in reporting my direction. Tan è =
Opposite Adjacent
= 5 (km / km leaves this unitless) = 0.5 10
è = tan -1 0.5 = 27E LOOK AT YOUR PICTURE! It’s 27E N of E [as opposed to 63E E of N–it is customary to report the “lesser” measure] ! !
All vectors have positive magnitude If a vector is NEGATIVE, the (!) refers to its direction
SUBTRACTION OF VECTORS !V means same magnitude, opposite direction
Kinematics in Two Dimensions & Vectors
2
MULTIPLICATION OF A VECTOR BY A SCALAR V × scalar quantity = magnified magnitude, same direction as before if scalar quantity is positive, opposite direction as before is scalar quantity is negative. ADDING VECTORS BY COMPONENTS We can “split” a vector into x-components and y-components and use a chart to solve for resultant vectors quite easily. What’s a component? Consider the vector V shown in diagram (a) below:
Start by drawing a set of axes at the “head” of the vector (its origin if you prefer). The x and y-components lie along their respective axes (b) and their magnitudes are equal to: Vy = V sin è Vx = V cos è You’ll come back to this method over and over between now and May! We’ll work some examples in a bit. For now, here’s more useful trig stuff with which to wow your geeky friends [or just me!] This Pythagorean Relationship also comes in handy from time to time: sin2 è + cos 2 è = 1
Law of sines:
A
c
B
b
a
C
Law of cosines: Both of these laws are handy when you have a triangle that is not a right triangle! Kinematics in Two Dimensions & Vectors
3
Example 1 A rural mail carrier leaves the post office and drives 22.0 km in a northerly direction to the next town. She then drives in a direction 60.0E south of east for 47.0 km to another town. What is her displacement from the post office?
Example 2 An airplane trip involves three legs, with two stopovers. The first leg is due east for 620 km; the second leg is southeast for 440 km; and the third leg is 53E south of west, for 550 km. What is the plane’s total displacement?
Kinematics in Two Dimensions & Vectors
4
PROJECTILE MOTION Tiger Woods hitting a golf ball, a thrown or batted baseball, kicked footballs, speeding bullets, and athletes doing the long jump or high jump are all examples of projectile motion. OBJECTS PROJECTED PARALLEL TO THE GROUND/SURFACE ! ! ! ! !
We will often ignore air resistance (be thankful!) We will only consider the motion after the projection (again be thankful!) Galileo was the first to analyze projectile motion. To keep it simple, we assume the motion begins at t = 0 and at the origin of an x,y coordinate system so xo = yo = 0 Treat the 2 components separately, and you can apply the kinematics equations from last chapter to each component
Let’s consider a tiny ball rolling off a table. It’s velocity vector at each instant is tangent to the path. !
Consider vy: Initially it’s zero, then gravity kicks in so now, it’s vy = !1/2 gt 2 (if you assign “up” as positive, then y + 8 and !9)
!
Consider vx: Initially it’s zero, it experiences no acceleration in the x-direction, therefore its velocity remains constant and vx = 0
!
Gravity prevails! An object projected horizontally will reach the ground in the same time as an object dropped vertically since the vertical motions are the same in each case. Dazzle your slacker friends with this new found knowledge!
OBJECTS PROJECTED AT AN UPWARD ANGLE Now you’ve gone and done it! You have introduced an initial vertical velocity component, vyo !
Because of the downward acceleration of gravity, vy continually decreases (slows) until the object reaches its highest point on its path, at which point vy = 0. Yep, you guessed it, now that sucker starts to accelerate downward
!
Good news! vx remains constant. Make peace with this concept. Now. Right now.
!
How can this be? (It know, it seems too easy and you are a suspicious lot by nature, huh?) The answer is: Simply because no horizontal force is acting on the projectile in the x-direction, velocity in the x-direction remains constant!
Kinematics in Two Dimensions & Vectors
5
SHALL WE JUMP OFF A CLIFF? !
Use the kinematic equations separately for each component, vertical and horizontal (see charts below)
!
Simplifications: ax = 0 ay = !g = ! 9.8 m/s2 Choose è relative to x-axis and: vxo = Vo cos è vyo = Vo sin è
Example 3 A movie stunt driver on a motorcycle speeds horizontally off a 50.0 m high cliff. How fast must the motorcycle leave the cliff-top if it is to land on level ground below, 90.0 m from the base of the cliff where the cameras are?
Kinematics in Two Dimensions & Vectors
6
Example 4 A football is kicked at an angle èo = 37.0E with a velocity of 20.0 m/s. Calculate a) the maximum height
b) the time of travel before the football hits the ground
c) how far away it hits the ground
d) the velocity vector at the maximum height
e) the acceleration vector at maximum height. Assume the ball leaves the foot at ground level.
Example 5 A child sits upright in a wagon which is moving to the right at constant speed as shown in the figure. The child extends her hand and throws an apple straight upward, while the wagon continues to travel forward at constant speed. If air resistance is neglected, where will the apple land?
Kinematics in Two Dimensions & Vectors
7
Example 6 A boy on a small hill aims his water-balloon slingshot horizontally, straight at a second boy hanging from a tree branch a distance d away. At the instant the water balloon is released, the second boy lets go and falls from the tree, hoping to avoid being hit. Show that he made the wrong move!
LEVEL RANGE FORMULA:
FOR USE WHEN y = yo ONLY! Where è is the angle made with the horizontal x-axis. You will always get 2 answers (roots) è and 90E ! è.
Example 8 Suppose the football in Ex. 4 was a punt and left the punter’s foot at a height of 1.00 m above the ground. How far did the football travel before hitting the ground? Set xo = 0 and yo = 0.
Kinematics in Two Dimensions & Vectors
8
PROJECTILE MOTION IS PARABOLIC That means y is a function of x an has the form y = ax ! bx2, where a & b are constants for any specific parabolic motion. Make friends with your graphing calculator... RELATIVE VELOCITY--The main reason we need vectors! Example 10 A boat’s speed in still water is 1.85 m/s. If the boat is to travel directly across a river whose current has speed 1.20 m/s, at what upstream angle must the boat head?
Example 11 The same boat now heads directly across the stream whose current is still 1.20 m/s. a) What is the velocity of the boat relative to the shore?
b) If the river is 110 m wide, how long will it take to cross and how far downstream will the boat be then?
Kinematics in Two Dimensions & Vectors
9
Example 12 A plane whose airspeed is 200 km/h heads due north. But a 100-km/h northeast wind (that is, coming from the northeast) suddenly begins to blow. What is the resulting velocity of the plane with respect to the ground?
Kinematics in Two Dimensions & Vectors
10
