Question

1. The treasure map in the figure gives the following directions to the buried treasure: "Start at the old oak tree, walk due north for 510 paces, then due east for                 150 paces. Dig." But when you arrive, you find an angry dragon just north of the tree. To avoid the dragon, you set off along the yellow brick road at an angle 60                 ^\circ east of north. After walking 350 paces you see an opening through the woods.

a. Which direction should you go to reach the treasure?

b. How far should you go to reach the treasure?

2. A supply plane needs to drop a package of food to scientists working on a glacier in Greenland. The plane flies 190m  above the glacier at a speed of 160m/s .

a. How far short of the target should it drop the package?

3. An old-fashioned single-play vinyl record rotates on a turntable at 48.0rpm . What are (a) the angular velocity in rad/s and (b) the period of the motion?

4. Your roommate is working on his bicycle and has the bike upside down. He spins the 70.0cm -diameter wheel, and you notice that a pebble stuck in the tread goes by three times every second.

a. What is the pebble's speed?

b. What is the pebble's acceleration?

Answer 1

Let due north be positive y-direction
Let due east be positive x-direction

Location that you want to arrive at is: (100, 550)

Using the alternate path:
X: 350 cos (90 - 60) = 303.10
Y: 350 sin (90 - 60) = 175

So to reach treasure, you will need to travel a distance of:
D^2 = (150 - 303.10)^2 + (510 - 175)^2
D = 368.32 paces

Direction:
Tan (theta) = (Y/X)
theta = arctan(Y/X)
theta = arctan(175 / 303.10)
theta = 30.00 North of West

2) The time it takes to fall ( not including air friction ) is

t= sqrt ( 2 X distance of fall )/ 9.8 )) = 6.22 sec

Traveling at 114 m/s you'd have to release the package

6.22 * 160 = 996.31 m before the target

3) 1 rpm = 0.104

48 rpm = 5.026 rad/s

period of motion = 2* pi/w

w= angular velocity

2 * 3.14/ 6.59 = 0.59

4) Frequency of the pebble is f = 3 Hz.

angular velocity ?= 2? f. = 6?
Linear speed or velocity v = r ? = [0.70/2]*6? = 6.59m/s
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Acceleration = v^2 /r = 6.59^2 / 0.35 = 124.08 m/s^2 toward the center of the wheel.