Vectors. Vector : quantity that has magnitude and direction Scalar : quantity that has magnitude only Example: 20 mi north (vector) vs. 20 mi (scalar) Arrows represent vectors graphically. (or A ). Magnitude = . (or A ). (arrow length proportional to vector magnitude). A. A. B. B.
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(arrow length proportional to vector magnitude)
Axand Ay are determined from trigonometry:
Ax = Acosq and Ay = Asinq
Ax(Ay) could be positive or negative depending on whether or not it points in the +x or –x (+y or –y) direction
A = Ax+ Ay
A = (Ax2 + Ay2)1/2
CQ2: A man entered a cave and walked 100 m north. He then made a sharp turn 150° to the west and walked 87 m straight ahead. How far is the man from where he entered the cave? (Note: sin30° = 0.50; cos30° = 0.87.)
Solution (details given in class):
(a) 185 N, q = 77.8° from x axis
(b) 185 N, q = 258° from x axis
(same as 1–D)
(same as 1–D)
CQ3: A weather balloon travels upward for 6 km while the wind blows it 10 km north and 8 km east. Approximately what is its final displacement from its initial position?
(Usually put start of motion at origin of coordinates)
ax = 0 vx = v0x = v0cosq (constant in time)
ay = –g vy varies with time
CQ4: Two skydivers are playing catch with a ball while they are falling through the air. Ignoring air resistance, in which direction should one skydiver throw the ball relative to the other if the one wants the other to catch it?
Velocity Components Interactive
Position and Time Interactive
y–direction (ay = –g):
x–direction (ax = 0):
x = xmax
What is the maximum height reached? (or At what value of y is vy = 0?)
What is the maximum range? (or What is the value of x when y = 0 [other than at the origin]?)Projectile Motion
A student stands at the edge of a cliff and throws a stone horizontally over the edge with a speed of 18.0 m/s. The cliff is 50.0 m above a flat, horizontal beach, as shown in the figure at right. (a) What are the coordinates of the initial position of the stone? (b) What are the components of the initial velocity? (c) Write the equations for the x- and y-components of the velocity of the stone with time. (d) Write the equations for the position of the stone with time. (e) How long after being released does the stone strike the beach below the cliff? (f) With what speed and angle of impact does the stone land?
Partial solution (details given in class):
(b) 1.78 s
(a) 32.5 mExample Problem #3.31
A car is parked on a cliff overlooking the ocean on an incline that makes an angle of 24.0° below the horizontal. The negligent driver leaves the car in neutral, and the emergency brakes are defective. The car rolls from rest down the incline with a constant acceleration of 4.00 m/s2 for a distance of 50.0 m to the edge of the cliff, which is 30.0 m above the ocean. Find (a) the car’s position relative to the base of the cliff when the car lands in the ocean, and (b) the length of time the car is in the air.
ActivPhysics Problem #3.7, Pearson/Addison Wesley (1995–2007)