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Kinematics in two dimensions
• Position
• Displacement
• Average Velocity
• (Instantaneous) Velocity
• Average Acceleration
• (Instantaneous) Acceleration
ˆ ˆr xi yj
2 1ˆ ˆr r r xi yj
rv
t
drv
dt
va
t
dva
dt
Instantaneous velocity and acceleration
x y z
dr dx dy dzˆ ˆ ˆ ˆ ˆ ˆv i j k v i v j v kdt dt dt dt
yx zx y z
dvdv dvdv ˆ ˆ ˆ ˆ ˆ ˆa i j k a i a j a kdt dt dt dt
Constant acceleration
0v v at 2
0 0
1x x v t at
2
1 - D
0v v at
20 0
1r r v t at
2
3 - D
x x0 xv v a t
20 x0 x
1x x v t a t
2
2 2x x0 x 0v v 2a x x
y y0 yv v a t
20 y0 y
1y y v t a t
2
2 2y y0 y 0v v 2a y y
0
0
0
ya g
Mortars, artillery, and guns
x0 0v v cos
y0 0v v sin
2 20 0
m
v siny
2g
2
0 0v sin2R
g
y0 22
x0 x0
v gy x x
v 2v
Motion is parabolic!
2004 Report on the status of the Army Artillery School
• ...And student officers with reading and math skills below seventh-grade level have been allowed to recycle multiple times before being force-fed to their units - even though an artillery officer who can't do math is like a fireman who's allergic to smoke.
• Columnist and former soldier
David H. Hackworth is the author of The Price of Honor, and contributes weekly commentary to DefenseWatch
Example Problem
• A batter hits a pitched ball whose center is 4.0 feet above the ground so that is angle of projection is 45 degrees and its range is 350 feet. The ball will be a home run if it clears a 24 ft high fence that is 320 feet from home plate. Will the ball clear the fence? If so, by how much?
Uniform Circular Motion Terms
• Frequency (f) – The number of revolutions (cycles) completed in a second.
• Period (T) – The time to complete one revolution (cycle).
1T
f
distance in one revolution 2 rv
time to complete one revolution T
Uniform Circular Motion Example
• A particle moves in a circular path 0.40 m in radius with constant speed. The particle makes 5.0 revolutions in each second of its motion.
• Find the particle’s speed
• Find the particle’s acceleration
13 m/s
390 m/s2
Relative Motion
BS BW WSv v v
The velocity of an object relative to stationary reference frame is equal to the velocity of an object relative to a moving reference frame plus the velocity of the moving reference frame.
Relative Motion
There is a 3 knot westerly current. Your ship is doing 10 knots by log. What course must you steer to make course north good?What will be your speed over ground?
017
9.5 kts