1. 3. Motion in 2- & 3-D Vectors Velocity & Acceleration Vectors
Relative Motion
Constant Acceleration
Projectile Motion
Uniform Circular Motion

2. At what angle should this penguin leave the water to maximize the range of its jump?
45

3. Vectors
Vectors:
Physics: Quantities with both magnitude & direction.
Mathematics : Members of a linear space.
(Free vectors)
Scalars: Quantities with only magnitude.
Displacement
Position vector
Vector addition:
Commutative: A + B = B + A
Associative: (A + B) + C = A +( B + C )

4. Multiplication by scalar.z
Coordinate system.
Cartesian coordinate system. A
Az k = Az y k j A Ay
= Ay j
Ax i = Ax
Ay j = Ay i y j  x x i Ax
= Ax i
Vector components:
Unit vectors:

5. Example 3.1. Taking a Drive
You drive to city 160 km from home, going 35 N of E.
Express your new position in unit vector notation, using an E-W / N-S coordinate system.
y (N)
city
r = 160 km j
 = 35
x (E)
home i

6. Vector Arithmetic with Unit Vectors

7. 3.2. Velocity & Acceleration Vectors
Average velocity
(Instantaneous) velocity
Average acceleration
(Instantaneous) acceleration

8. Velocity & Acceleration in 2-D
a  v  circular motion

9. 3.3. Relative Motion Motion is relative (requires frame of reference).
Man walks at v = 4 km/h down aisle to front of plane,
which move at V = 1000 km/h wrt (with respect to) ground.
Man’s velocity wrt ground is v = v + V.
Plane flies at v wrt air.
Air moves at V wrt ground.
Plane’s velocity wrt ground is v = v + V.

10. Example 3.2. Navigating a Jetliner
Jet flies at 960 km / h wrt air, trying to reach airport km northward.
Assuming wind blows steadly eastward at 190 km / h.
What direction should the plane fly?
How long will the trip takes?
Desired velocity
Wind velocity
V 190 km/h
Jet velocity  v v
960 km/h 
Trip time

11. 3.4. Constant Acceleration
2-D:

12. Example 3.3. Windsurfing net displacement
You’re windsurfing at 7.3 m/s when a wind gust accelerates you
at m/s2 at 60 to your original direction.
If the gust lasts 8.7 s, what is your net displacement?
net displacement

13. 3.5. Projectile Motion
2-D motion under constant gravitational acceleration
parabola

14. Example Washout
A section of highway was washed away by flood, creating a gash 1.7 m deep.
A car moving at 31 m/s goes over the edge.
How far from the edge does it land?

15. Projectile Trajectory
Projectile trajectory:
parabola

16. Example 3.5. Out of the Hole   Lands at 5.5 m from edge.
A construction worker stands in a 2.6 m deep hole, 3.1 m from edge of hole.
He tosses a hammer to a companion outside the hole.
Let the hammer leave his hand 1.0 m above hole bottom at an angle of 35.
What’s the minimum speed for it to clear the edge?
How far from the edge does it land?
minimum speed 
 Lands at 5.5 m from edge.

17. The Range of a Projectile
Horizontal range y = y0 : 
Longest range at 0 = 45 = /4.
Prob 70: Range is same for 0 & /2  0.
Prob 2.77: Projectile spends 71% in upper half of trajectory.

18. Example 3.6. Probing the Atmosphere
After a short engine firing, a rocket reaches 4.6 km/s.
If the rocket is to land within 50 km from its launch site,
what’s the maximum allowable deviation from a vertical trajectory?
Short engine firing  y  0, v0 = 4.6 km/s. 
 maximum allowable deviation from a vertical trajectory is 0.67.

19. 3.6. Uniform Circular Motion
Uniform circular motion: circular trajectory, constant speed.
Examples:
Satellite orbit.
Planetary orbits (almost).
Earth’s rotation.
Motors.
Electrons in magnetic field. ⁞

20.  
( centripetal )

21. Example 3.7. Space Shuttle Orbit
Orbit of space shuttle is circular at altitude 250 km, where g is 93% of its surface value.
Find its orbital period.
(low orbits)
ISS: r ~ 350 km
15.7 orbits a day

22. Example 3.7. Engineering a Road
Consider a flat, horizontal road with 80 km/h (22.2 m/s) speed limit.
If the max vehicle acceleration is 1.5 m/s2,
what’s the min safe radius for curves on this road.

23. Nonuniform Circular Motion
Nonuniform Circular Motion: trajectory circular, speed nonuniform
 a non-radial but ar = v2 / r v at ar a

24. GOT IT? 3.4. Arbitrary motion: ar = v2 / r r = radius of curvature
If v1 = v4 , & v2 = v3 , rank ak.
Ans: a2 > a3 > a4 > a1