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

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

3.1. 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 )

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:

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

Vector Arithmetic with Unit Vectors

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

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

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.

Example 3.2. Navigating a Jetliner Jet flies at 960 km / h wrt air, trying to reach airport 1290 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

3.4. Constant Acceleration 2-D:

Example 3.3. Windsurfing net displacement You’re windsurfing at 7.3 m/s when a wind gust accelerates you at 0.82 m/s2 at 60 to your original direction. If the gust lasts 8.7 s, what is your net displacement? net displacement

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

Example 3.4. 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?

Projectile Trajectory  Projectile trajectory: parabola

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.

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.

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.

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. ⁞

  ( centripetal )

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

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.

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

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