Vectors & 2D Kinematics (W & R Recit. only!)

PHYS 250 Recitation 04: Vectors + 2D Kinematics
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Position, displacement, velocity, acceleration and force are all vectors, which
means that they have both magnitude and direction. One can represent any vector
as an arrow whose length indicates the magnitude (and whose direction, obviously,
represents the direction of the vector).
v
v x
v y
This magnitude-angle representation (e.g., v and for a velocity vector) is useful
for visualizing vectors, but for doing calculations we usually use component
representation (e.g., v x and v y for a velocity vector). To convert between these two
representations, we use trigonometry and the Pythagorean theorem:
v x = v cos() v 2 = v 2 2
x + v y
v y = v sin()
tan() = v y /v x
(where is the angle the vector makes from the +x-axis, and towards the y-axis is positive)
Vectors can be added or subtracted. Graphically, you add vectors tip to tail. In
terms of components, you add the components (i.e., add the x-components to get
the x-component of the sum; add the y-components to get the y-components of the
sum). To subtract vectors, just think of it as: C = A – B = A + (-B), where –B is
just B flipped (i.e., rotated by 180 degrees).
As you will see in lecture, any 2D problem can be viewed as two independent 1D
problems. For each direction (e.g., horizontal and vertical), the x-v x -a x slope & area
relationships and the X/V/V 2 equations (if constant acceleration!) hold. Before
doing anything else in a 2D kinematics problem, decompose all magnitude-angle
vectors into components. Then it’s basically just like doing 1D kinematics
problems like you did in last week’s recitation. If your final answer is a vector with
both x- and y-components, you may need to express your final answer in
magnitude-angle representation.

Problem #1. Manipulate vectors A & B below as arrows to draw the solution to the
following expressions:
A
B
a) 2A + B =
b) A – B =
c) Draw C, where A + B + C = 0

Problem #2. Manipulate vectors A & B below in terms of components to calculate
the following quantities:
A = 20 m/s at a 30 degree angle above the x-axis
B = 10 m/s at a 40 degree angle left of the y-axis
a) What are A and B expressed in terms of x- and y-components?
a) A/2 + B =
b) A – 3B =
c) Find the components of C, where A + B + C = 0
d) What is C in terms of magnitude & angle above the x-axis?

Problem #3: A ball is thrown at 20 m/s at an angle of 30 degrees up above the
horizontal. The only thing acting on the ball is gravity, so the ball has an
acceleration of 9.8 m/s 2 in the downwards (-y) direction. What is the velocity of the
ball two seconds later? (Assume that it is still in the air at 2 seconds.)
First identify all relevant horizontal and vertical quantities (given/known/implied
or asked for), and those quantities that need to be decomposed into x- and y-
components.
Next, find the vector components, and finally solve the problem (as two separate
1D problems). You can leave your answer in component form.

Problem #4: A penguin sliding on ice has an initial velocity of 5 m/s due north.
Due to a strong wind, the penguin has an acceleration of 2 m/s 2 30 degrees west of
North. What is the penguin’s displacement (magnitude and direction) after 4
seconds?
First identify all relevant x- (East-West) and y- (North-South) quantities
(given/known/implied or asked for), and those quantities that need to be
decomposed into x- and y-components.
Next, find the vector components, and finally solve the problem (as two separate
1D problems). Solve in component form and then convert your answer to
magnitude-angle.