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Topic 1.3 Extended B - Components of motion
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Up to now we have considered objects moving in one dimension. However, most objects move in more than one dimension.For example, consider the ball shown here:
Motion in Two Dimensions3-1 Components of motion
We can sketch in our x and y for successive snapshots to obtain an idea of the different velocities the ball has at different times: x is in YELLOW. y is in RED. We can also sketch in the displacement d of the ball at each time interval (in GREEN).Let's examine one time interval in detail:
x
y
d
y
FYI: The displacement vector gives the direction of the motion
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From the Pythagorean Theorem we can find the value of d if we know x and y:d2 = x2 + y2
Topic 1.3 Extended B - Components of motion
x
y
d
y
d = x2 + y2 Magnitude of a 2D displacement
If we know the time interval t between snapshots, we can find the velocity of the ball simply by dividing the displacements shown above by t. The proportions of our triangle will not change.
vx
vy v
vy
Thus
v = vx2 + vy
2 Magnitude of a
2D velocity
Each triangle gets a good name:
displacement triangle
velocity triangle
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We call the vx the horizontal component of the velocity.
Topic 1.3 Extended B - Components of motion
vx
vy v
vy
horizontal component
We call the vy the vertical component of the velocity.
vertical
component
From trigonometry we know there is a relationship between the sides of a triangle, and the angle :
opphyp
adjhyp
oppadj
hypotenuse
adjacentopposite
θ
trigonometric ratios
s-o-h-c-a-h-t-o-a
v
vx = v cos θ
vy = v sin θ
v
vxvy
vsin θ = cos θ = tan θ =
vx
vy
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Suppose we know the velocity of the ball is 25.0 m/s at an angle of 30° with respect to (wrt) the positive x-axis.
Topic 1.3 Extended B - Components of motion
vx
vy v
vy
What is vx the horizontal component of the velocity?
vx = v cos θ
vy = v sin θ
vx = v cos θ
vx = (25.0 m/s)cos 30°
vx = 21.7 m/s
What is vy the vertical component of the velocity?
vy = v sin θ
vy = (25.0 m/s)sin 30°
vy = 12.5 m/sFYI: You can check your results by squaring each answer, summing, and taking the square root. What should you get?
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Sometimes we know the components of the velocity, and want to find the magnitude and the direction:
Topic 1.3 Extended B - Components of motion
vx
vy v
vy
Suppose vx = 30.0 m/s.
Suppose vy = 40.0 m/s.
Thenv = vx
2 + vy2
v = 302 + 402
v = 50.0 m/s magnitude
of v
oppadj
tan θ =vx
vy=40 m/s
=30 m/s
and
so that
θ = tan-1 43
= 53.1°direction
of v
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Sometimes we know the formulas for the components of the velocity of a ball, and want to find the magnitude and the direction of the velocity at a particular time:
Topic 1.3 Extended B - Components of motion
Suppose vx = 30.0 (measured in m/s).Suppose vy = 40.0 - 5t (vy in m/s, t in s)Then what is the velocity at t = 2 s?
v = vx2 + vy
2
v = 302 + 302
v = 42.4 m/s magnitude
of v
oppadj
tan θ =vx
vy=30 m/s
=30 m/s
What is the direction of the ball at this instant?
so that
θ = tan-1(1) = 45.0°direction
of v
vx = 30.0 m/s
vy = 40 - 5(2)
vy = 30.0 m/s