1. Physics 103: Lecture 5 2D Motion + Relative Velocities
Today’s lecture will be on
More on 2D motion
Addition of velocities
Velocity (like position and acceleration) is a vector.
Recall a vector is a quantity with both direction and magnitude.
A vector can be resolved into components (usually the axes of an orthogonal coordinate system)
(NOT unique -- depends on the coordinate system.
Addition of velocities is vector addition NOT algebraic addition.
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Physics 103, Spring 2004, U.Wisconsin 1

2. Summary of Lecture 4 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
Choose alignment of the coordinate system to simplify the problem. (Choice always allows simplification!)
For kinematic problems this means one axis along the vector acceleration.
Often this is gravity ie two axes are horizontal and vertical.
x and y motions are independent!
They share a common time t
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Physics 103, Spring 2004, U.Wisconsin

3. 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
There is no acceleration along horizontal - object continues to travel at constant speed (same as that of the plane) along horizontal. Due to gravitational acceleration the object’s speed downwards increases.
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Physics 103, Spring 2004, U.Wisconsin

4. 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

5. Physics 103, Spring 2004, U.Wisconsin
Lecture 5, Pre-Flight 1&2
You are a vet trying to shoot a tranquilizer dart into a monkey hanging from a branch in a distant tree. You know that the monkey is very nervous, and will let go of the branch and start to fall as soon as your gun goes off. On the other hand, you also know that the dart will not travel in a straight line, but rather in a parabolic path like any other projectile. In order to hit the monkey with the dart, where should you point the gun before shooting?
1 Right at the monkey
2 Below the monkey
3 Above the monkey
correct
If the shot is fired at the monkey the same time the monkey drops, both objects will fall at the same rate causing the shot to hit the monkey.
since the monkey is going to start falling right away you need to aim below it
Along the way, gravity is going to pull the dart down, so you would have to aim up. Aiming right at it or below would miss the monkey by going underneath it.
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Physics 103, Spring 2004, U.Wisconsin

6. Physics 103, Spring 2004, U.Wisconsin
Shooting the Monkey...
x = v0 t
y = -1/2 g t2
x = x0
y = -1/2 g t2
Dart hits the
monkey!
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Physics 103, Spring 2004, U.Wisconsin

7. Physics 103, Spring 2004, U.Wisconsin
Shooting the Monkey...
r = r0 - 1/2 g t2
At an angle, still aim at the monkey!
Dart hits the
monkey!
r = v0 t - 1/2 g t2
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Physics 103, Spring 2004, U.Wisconsin

8. Shooting the Enemy Paratrooper...
If a soldier wants to shoot down an enemy paratrooper descending at uniform speed, Se, where should he aim?
Above the enemy
At the enemy
Below the enemy
Answer depends on the enemy’s position and vertical speed.
r = r0 - Se t
r = v0 t - 1/2 g t2
Miss the enemy
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Physics 103, Spring 2004, U.Wisconsin

9. Reference Frames: Relative Motion
VCB=VCA+VAB
Velocity of B relative to ground ( C ) : VCB
Velocity of A relative to ground ( C ) : VCA
Velocity of B relative to A : VAB
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Physics 103, Spring 2004, U.Wisconsin

10. Physics 103, Spring 2004, U.Wisconsin
Relative Motion
If an airplane flies in a jet stream, depending on the relative orientation of the airplane and the jet stream, the plane can go faster or slower than it normally would in the absence of the jet stream
If a person rows a boat across a rapidly flowing river and tries to head directly for the shore, the boat moves diagonally relative to the shore
Velocity is a vector - add velocities like vectors
Sum the components
Vx = V1x + V2x
V1x = V1 cosq
Vy = V1y + V2y
V1x = V1 sinq
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Physics 103, Spring 2004, U.Wisconsin

11. Physics 103, Spring 2004, U.Wisconsin
Lecture 5, Pre-Flight 3&4
Three swimmers can swim equally fast relative to the water. They have a race to see who can swim across a river in the least time. Relative to the water, Beth (B) swims perpendicular to the flow, Ann (A) swims upstream, and Carly (C) swims downstream. Which swimmer wins the race?
A) Ann
B) Beth
C) Carly
correct
A B C
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Physics 103, Spring 2004, U.Wisconsin

12. Lecture 5, Pre-Flight 3&4 (great answers)
Beth will reach the shore first because the vertical component of her velocity is greater than that of the other swimmers.
The key here is how fast the vector in the vertical direction is. "B" focuses all of its speed on the vertical vector, while the others divert some of their speed to the horizontal vectors.
A B C
Time to get across =
width of river/vertical component of velocity
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Physics 103, Spring 2004, U.Wisconsin

13. Lecture 5, Pre-Flight 3&4 (common misconceptions)
While Carly is moving forward she will also be moving along with the current. two positive(+) direction motions = faster velocity.
Carly will get there first because she is using the current to her advantage.
A B C
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Physics 103, Spring 2004, U.Wisconsin

14. Physics 103, Spring 2004, U.Wisconsin
Followup Question
Heather wants to swim across a flowing river in such a way that she ends up on the opposite side directly opposite her starting point. She should therefore aim….
1) upstream
2) downstream
3) directly across
correct
vsg
vwg
vsw
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Physics 103, Spring 2004, U.Wisconsin

15. Lecture 5, Pre-Flight 5 and 6
A seagull flies through the air with a velocity of 9 m/s if there were no wind. However, it is making the same effort and flying in a headwind. If it takes the bird 20 minutes to travel 6 km as measured on the earth, what is the velocity of the wind?
1. 4 m/s
2. -4 m/s
3. 13 m/s
m/s
correct
Seagull’s velocity in the reference frame of the wind = 9 m/s
i. e., in this frame, wind velocity is zero
Seagull travels at 6000/1200 = 5 m/s relative to earth. Therefore, the wind velocity relative to earth is 5-9=-4 m/s.
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Physics 103, Spring 2004, U.Wisconsin

16. Follow-up, Pre-Flight 5 and 6
If the seagull turns around and flies back how long will it take to return?
1. More time than for flying out
2. Less time than for flying out
3. The same amount of time
correct
Seagull’s return velocity is: -4-9=-13 m/s. The speed is higher so it takes less time to return. Time taken for the return is given by 6000 m / 13 (m/s) = s = 461.5/60 = 7.69 minutes
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Physics 103, Spring 2004, U.Wisconsin

17. Follow-up 2, Pre-Flight 5 and 6
How are the round-trip times with and without wind related if the seagull always goes at 9 m/s?
1. The round-trip time is the same with/without the wind
2. The round trip time is always larger with the wind
3. It is not possible to calculate this
correct
Time taken for the round trip with wind is: minutes
Time taken for the round trip without wind is: m / 9 m/s = 1333 s = 22.2 minutes
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Physics 103, Spring 2004, U.Wisconsin

18. Physics 103, Spring 2004, U.Wisconsin
Once again:
A boat is drifting in a river which has a current of 1 mph. The boat is a half mile upstream of a rock when an observer on the boat sees a seagull overhead. The observer sees the gull flying continuously back and forth at constant speed (10 mph) between the boat and the rock. When the boat passes close by the rock, how far (what distance) has the gull flown?
5 miles
10 miles
Not sufficient information
to determine the distance
correct
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Physics 103, Spring 2004, U.Wisconsin

19. Pendulum Motion - 2 Dimensions
Kinematic equations for constant acceleration
DO NOT APPLY for this case.
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Physics 103, Spring 2004, U.Wisconsin

20. Summary Relative Motion
Velocity of seagull with respect to ground, Vsg
Velocity of seagull with respect to wind, Vsw (i.e., velocity with which it would fly in calm winds)
Velocity of wind with respect to the ground, Vwg
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Physics 103, Spring 2004, U.Wisconsin