1. Physics 103: Lecture 4 Vectors - Motion in Two Dimensions
Today’s lecture will be on
Vectors
Two dimensions
projectile motion
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Physics 103, Spring 2004, U.Wisconsin 1

2. Physics 103, Spring 2004, U.Wisconsin
One Dimension }
Define origin
Define sense of direction
Position is a signed number (direction and magnitude)
Displacement, velocity, acceleration are also vectors specified just by signed numbers
Reference Frame
…-4 -3 -2 -1 1 2 3 4…
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Physics 103, Spring 2004, U.Wisconsin

3. Physics 103, Spring 2004, U.Wisconsin
Two Dimensions
Position can be anywhere in the plane
Again, select an origin
Draw two mutually perpendicular lines meeting at the origin
Select +/- directions for horizontal (x) and vertical (y) axes
Any position in the plane is given by two signed numbers
A vector points to this position
The square of its length is, R2= Rx 2+ Ry 2
The angle of that vector is,  = tan-1(Ry / Rx) R Ry
Rx = R cos 
Ry = R sin  q Rx
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Physics 103, Spring 2004, U.Wisconsin

4. Physics 103, Spring 2004, U.Wisconsin
Preflight 4, Q 1 & 2
Can a vector have a component bigger than its magnitude?
Yes No
The square of magnitude of a vector is given in terms of its components by R2= Rx 2+ Ry 2
Since the square is always positive the components cannot be larger than the magnitude
9/17/2018
Physics 103, Spring 2004, U.Wisconsin

5. Physics 103, Spring 2004, U.Wisconsin
Preflight 4, Q 3 & 4
The sum of the two components of a non-zero 2-D vector is zero. Which of these directions is the vector pointing in?
45o
90o
135o
180o
135o
-45o
The sum of components is zero implies Rx = - Ry
The angle,  = tan-1(Ry / Rx) = tan-1 -1 = 135o = -45o (not unique, ± multiples of 2 )
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Physics 103, Spring 2004, U.Wisconsin

6. Physics 103, Spring 2004, U.Wisconsin
Vector Algebra
Analytical method
Add the components separately to get the components of sum vector
Rx = R1x + R2x
Ry = R1y + R2y
Scalar multiplication of vector
Can change magnitude and sign
Multiply all components by scalar
Components of sR are sRx and sRy
Negation of vector (multiplying by -1)
Reverse signs of both components
Vector points in opposite direction
R=R1+R2
D=R2-R1 y R1 R2 D x
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Physics 103, Spring 2004, U.Wisconsin

7. Physics 103, Spring 2004, U.Wisconsin
Vectors!
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Physics 103, Spring 2004, U.Wisconsin

8. Summary of Lecture 3 Free-fall: ay = -g = -9.81 m/s2
Equations with constant acceleration
x = v0t + 1/2 at2
v = at
v2 = v02 + 2a x
Free-fall: ay = -g = m/s2
y = y0 + v0yt - 1/2 gt2
vy = v0y - gt
vy2 = v0y2 - 2gy
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Physics 103, Spring 2004, U.Wisconsin

9. Two Dimensional Motion
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Physics 103, Spring 2004, U.Wisconsin

10. Range of Soccer Ball Dimensional Analysis: Strategy: 9/17/2018
Physics 103, Spring 2004, U.Wisconsin

11. Kinematics in Two Dimensions
x = x0 + v0xt + 1/2 axt2
vx = v0x + axt
vx2 = v0x2 + 2ax x
y = y0 + v0yt + 1/2 ayt2
vy = v0y + ayt
vy2 = v0y2 + 2ay y
x and y motions are independent!
They share a common time t
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Physics 103, Spring 2004, U.Wisconsin

12. Kinematics for Projectile Motion ax = 0 ay = -g
y = y0 + v0yt - 1/2 gt2
vy = v0y - gt
vy2 = v0y2 - 2g y
x = x0 + vxt
vx = v0x
x and y motions are independent!
They share a common time t
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Physics 103, Spring 2004, U.Wisconsin

13. Projectile Motion: Maximum height reached Time taken for getting there
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Physics 103, Spring 2004, U.Wisconsin

14. Projectile Motion: Maximum Range
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Physics 103, Spring 2004, U.Wisconsin

15. Physics 103, Spring 2004, U.Wisconsin
Soccer Ball
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Physics 103, Spring 2004, U.Wisconsin

16. Higher the shell flies, the longer it takes.
A battleship simultaneously fires two shells at enemy ships from identical canons. If the shells follow the parabolic trajectories shown, which ship gets hit first?
1. Ship A.
2. Ship B.
3. Both at the same time
Higher the shell flies, the longer it takes.
What should the captain order if he wants to hit both ships at the same time? A B
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Physics 103, Spring 2004, U.Wisconsin

17. Physics 103, Spring 2004, U.Wisconsin
Lecture 4, Pre-Flight 5&6
You and a friend are standing on level ground, each holding identical baseballs. At exactly the same time, and from the same height, you drop your baseball without throwing it while your friend throws her baseball horizontally as hard as she can. Which ball hits the ground first?
1. Your ball
2. Your friends ball
3. They both hit the ground at the same time
correct
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Physics 103, Spring 2004, U.Wisconsin

18. Lecture 4, Pre-Flight 5&6 (great answers)
They both have the same initial vertical component with the same acceleration due to gravity, therefore they hit the ground at the same time.
No matter how much horizontal velocity is put on an object it still falls at the same rate as any other dropped object.
y = y0 + voyt - gt2/2
v0y = 0 and y=0
Therefore, t=sqrt(2y0/g)
Result is independent of v0x
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Physics 103, Spring 2004, U.Wisconsin

19. Physics 103, Spring 2004, U.Wisconsin
Question?
Without air resistance, an object dropped from a plane flying at constant speed in a straight line will
1. Quickly lag behind the plane.
2. Remain vertically under the plane.
3. Move ahead of the plane
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Physics 103, Spring 2004, U.Wisconsin

20. Physics 103, Spring 2004, U.Wisconsin
Lecture 4, Pre-Flight 7&8
A flatbed railroad car is moving along a track at constant velocity. A passenger at the center of the car throws a ball straight up. Neglecting air resistance, where will the ball land?
1. Forward of the center of the car
2. At the center of the car
3. Backward of the center of the car
correct
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Physics 103, Spring 2004, U.Wisconsin

21. Physics 103, Spring 2004, U.Wisconsin
Great Answers!
The train and the ball have the same horizontal velocity and by throwing the ball straight up, the horizontal component is not changed.
The ball has no acceleration in the horizontal direction. Therefore, the balls
remains directly above the center of the train at all times during the flight
and would fall directly back toward the center of the train.
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Physics 103, Spring 2004, U.Wisconsin

22. Physics 103, Spring 2004, U.Wisconsin
Summary
Projectile Motion
y = y0 + v0yt - 1/2 gt2
vy = v0y - gt
vy2 = v0y2 - 2g y
x = x0 + v0t
v = v0x
9/17/2018
Physics 103, Spring 2004, U.Wisconsin