Kinematics in two dimensions
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Transcript of Kinematics in two dimensions
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
Projectile Motionx x0v v
0 x0x x v t
y y0v v gt
20 y0
1y y v t gt
2
2 2y y0 0v v 2g y y
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
• Uniform = magnitude of velocity (speed) is not changing
2
R
va
r
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
Mo Boards
0
3 min
1000 yds
090/15kts
135/10kts
BS B you you Sv v v