Chapter 4: TwoDimensional Kinematics

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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.
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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?

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Motion in 2-dimensions with constant acceleration.
Horizontal (x)
Vertical (y)
x(t) x-coordinate at time t x0 x-coordinate
at time tt0 x(t0) vx(t) velocity component
in x-direction at time t . vx,0 velocity
component in x-direction at time tt0
y(t) y-coordinate at time t y0 x-coordinate
at time tt0 y(t0) vy(t) velocity component
in y-direction at time t . vy,0 velocity
component in y-direction at time tt0
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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
• The Acceleration of gravity means the
  acceleration of an object that is in free fall,
  with no other important forces (see chapter 5).
• Horizontal velocity is constant ax 0
• Vertical motion governed by constant acceleration
  of gravity

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Free Fall (up y)
Horizontal (x)
Vertical (y)
ax 0
ay -g -9.80 m/s2 -32.0 ft/s2 g 9.80 m/s2
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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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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

• Notice that y vs x graph looks like y vs t graph
• 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)?

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General Launch Angle
Consider an object launched from the origin at an
angle q with respect to the horizontal.
Find the x and y components of the initial
velocity vector, in terms of the angle of launch
q and the initial velocity v0.
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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?
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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.
0 v0,y-g tHangTime /2 tHangTime 2 v0 sinq/g
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Quiz 4.1
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 horizontal component of initial
velocity
(smallest to largest)

• (a, b,c)
• (c, b, a)
• (b, a, c) or (b, c, a)
• (a, c, b) or (c, a, b)

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Quiz 4.2
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 time in flight (hang time).
(smallest to largest)

• (a, b,c)
• (c, b, a)
• (b, a, c) or (b, c, a)
• (a, c, b) or (c, a, b)

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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 (hang time).
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Range (unequal heights)
Set y0 and solve quadratic for t
What is physical meaning of ?sign?
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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?
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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.