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Chapter 4: TwoDimensional Kinematics

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Title: Chapter 4: TwoDimensional Kinematics


1
Chapter 4 Two-Dimensional Kinematics
Motion in the x direction is independent from
motion in the y direction. We use the same
equations from Chapter 2, but for each dimension
separately.
There are not really any new equations in this
chapter.
2
Example
  • A car drives in a straight line with a constant
    speed of 20 m/s in a direction 30 East of North.
  • How long will it take the car to travel 60 m East
    of its starting location?
  • How far North will the car have gone in that
    length of time?

3
Projectile Motion
  • Assume that acceleration of gravity is constant,
    downward and has a magnitude of g 9.81 m/s2
  • Air resistance is ignored
  • The Earths rotation is ignored
  • The variation of gravity as a function of
    distance from the center of the earth is ignored
  • Horizontal velocity is constant ax 0
  • Vertical motion governed by constant acceleration
    of gravity

4
Motion of a Projectile Launched Horizontally
The dots represents the position of the object
every 0.05 s.
t 1 s
y0 8 m x0 0 v0y 0
v0x 6 m/s ay -9.81 m/s2 ax 0
  • Verify the position of the object at t 1s.
  • What would be the position of the object at t 1
    s if it were dropped (v0x 0)?

5
Walker, Problem 11, pg. 99
  • Pitchers mounds are raised to compensate for the
    vertical drop of the ball as it travels 18 m to
    the catcher.
  • If a pitch is thrown horizontally with an initial
    speed of 32 m/s, how far does it drop by the time
    it reaches the catcher?
  • If the speed of the pitch is increased, does the
    drop distance increase, decrease or stay the
    same? Explain.
  • If this baseball game were to be played on the
    moon, would the drop distance increase, decrease,
    or stay the same? Explain.

6
General Launch Angle
Consider an object launched from the origin at an
angle q with respect to the horizontal.
Fine the x and y components of the initial
velocity vector.
7
Walker, Problem 37, pg. 101
On a hot summer day a young girl swings on a rope
above the local swimming hole. When she lets go
of the rope her initial velocity is 2.25 m/s at
an angle of 35.0 above the horizontal. If she
is in flight for 1.60 s, how high above the water
was she when she let go of the rope?
What is the girls minimum speed during her
flight? What is her acceleration at the top of
her trajectory?
8
Range
The range R of a projectile is the horizontal
distance it travels before landing.
assuming same initial and final elevation
What angle q results in the maximum range?
What if we do not ignore air resistance?
9
Range (unequal heights)
Set y0 and solve quadratic for t
10
Maximum Height
The maximum height (and therefore the hang
time) of a projectile depends only on the
vertical component of its initial velocity.
At ymax, the vertical velocity vy is zero.
11
Walker CQ 11, pg. 98
Three projectiles (a, b and c) are launched with
the same initial speed but with different launch
angles, as shown. List the projectiles in order
of increasing (a) horizontal component of initial
velocity and (b) time in flight
12
Walker, CQ 12, pg. 98
Three projectiles (a, b and c) are launched with
different initial speeds so that they reach the
same maximum height, as shown. List the
projectiles in order of increasing (a) initial
speed and (b) time of flight.
13
Walker, CQ 14, pg. 98
Driving down the highway you find yourself behind
a heavily loaded tomato truck. You follow close
behind the truck, keeping the same speed.
Suddenly a tomato falls from the back of the
truck. Will the tomato hit your car or land on
the road, assuming you continue moving with the
same speed and direction? (Neglect air friction)
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