Name ___________________________________
1-dimensional
kinematics
vx =
vx =
dx
dt
Δx
Δt
ax =
ax =
dv x
dt
Δv x
Δt
2-D or 3-D kinematics
dr dv
v=
a=
dt
dt
Δr
Δv
v avg =
a avg =
Δt
Δt
Relative Velocity
!
!
!
rP/A = rP/B + rB/A
!
!
!
vP/A = vP/B + vB/A
!
!
!
aP/A = aP/B + aB/A
g = 9.80 m/s2
Constant Acceleration
Projectile motion (with y positive up)
x = xo + voxt + ½ axt2
vx = vox
vx = vox + axt
x = xo + voxt
vx2 = vox2 + 2ax(x-xo)
x = xo+½ (vx+vox)t
y = yo + voyt –
1 2
gt
2
x = xo + vxt – ½ axt2
vy = voy – gt
Constant acceleration
vy 2 = voy2 − 2g(y − yo )
y = yo + voyt + ½ ayt2
vy = voy + ayt
vy2 = voy2 + 2ay(y-yo)
y = yo+½ (vy+voy)t
y = yo + vyt – ½ ayt2
?!”#$%$
4
= ??&
3
For the general quadratic equation 0 = Ax 2 + Bx + C ,
y = yo +
(v
y
+ voy )
t
2
y = yo + vyt +
1 2
gt
2
Uniform Circular Motion
?!”# = ? $ /?
?=
2??
?
?!”# = 4? $ ?⁄? $
x=
− B ± B 2 − 4AC
2A
g
Law of Cosines: C2 = A2 + B2 – 2AB cos g
A
Law of Sines:
sin α sin β sin γ
=
=
A
B
C
a
b
C
Page 1 of 9
B
Name ___________________________________
•
Whenever possible, solve equations using algebraic symbols. Plug in numbers at
the end!
•
To receive full credit, your line of reasoning must be clear. Show the formulas used,
the numbers you plug into the formulae, and correct units.
•
Assume all numerical values have 3 significant figures, and express final answers
to 3 significant figures.
•
Partial credit will be given for partial answers if your work communicates your
understanding of the problem.
•
Credit will not be given for answers without work shown.
•
Sharing of this document online or in any other manner without the explicit written
permission of its author, Prof. Matthew Searle, is strictly prohibited.
Page 2 of 9
Name ___________________________________
•
Physics 140
Exam 1
1.
a. (5 points) Determine the magnitude and direction (as measured from the +x
axis) of vector B. Show all work. Express your final answer in decimal form.
!
B
! = 5.10î − 3.80 ĵ
b. (5 points) Express the following vector in unit vector notation. Show all work and
express final answers in decimal form.
A = 4.70 m @ 280°
c. (5 points) Carefully sketch the vector sum and difference. Make sure you show
the resultant vector. Use the vectors A and B shown below. Note: These are
not the same A & B vectors from above. Also remember that you can pick up
vectors and drag them around, as long as you don’t change their magnitude or
direction.
!
!
!
!
a. !A + 2B
b. !− A + 2B
Page 3 of 9
Name ___________________________________
2. (10 points) An Earthlike Planet. In January 2006 astronomers reported the
discovery of a planet comparable in size to the earth orbiting another star and
having a mass about 7.60 times the earth’s mass. It is believed to consist of a
mixture of rock and ice, similar to Neptune. If this planet has the same density as
Neptune (1.38 g/cm^3), what is its radius expressed in kilometers?
Astronomical data:
Earth: Mass: 5.97×10^24 kg
Remember: Density = mass/volume
Page 4 of 9
Name ___________________________________
3. 3. (15 points) Bender, Leela and Fry are on a treasure hunt. Bender stole a
treasure map from a little kid, and they only have a short time to find it before the
kid comes back with his parents. Each of them have a meter stick, compass,
calculator, and shovel, and scratch down the following displacements in random
orders:
A
= 73.4 m, 31.0° north of east;
!
C
=19.9 m, due south;
!
B = 42.7 m, 28.0° south of west;
!
!
Draw a graph of A + B + C ! Remember vectors are added (+) head to tail. Do
NOT draw all vectors with tails at origin, but you may add them together in any
order you wish.
North = +y axis, East = +x axis, South = -y axis, West = -x axis
The three displacements lead to where a sweet bag of loot is buried. Leela and Fry
start measuring immediately, but Bender, using his mighty Robo-logic, first
calculates where to go. What resultant vector does he calculate? (Give your answer
in both coordinate and magnitude-angle notation).
Page 5 of 9
Name ___________________________________
4. (10 points) A motorcyclist heading east (+x direction) through a small town
accelerates after she passes the signpost for the outer limits of the city. Her
acceleration is a constant 3.15 m/s^2. At time t = 0 s, she is 3.50 m east of the
signpost, moving east at 12.5 m/s.
a) Find her position ( signpost is at x = 0), and velocity at time t = 2.75 s
b) Where (i.e. what’s her position?) is the motorcyclist when her velocity is 22.5
m/s?
Page 6 of 9
Name ___________________________________
5. (10 points) Cross-product & Dot product. Let vector A = 1i – 2j – 3k. Let vector B
= 2i -1j + 2k.
a) Calculate the dot product of the vectors using the component method. Then
use this result to find the angle between the vectors using the other formula for
the dot product.
c) Draw a xyz-coordinate system and the vectors A & B. Then, using the
determinant method, calculate the cross product AxB.
d) Then draw vector C = A x B.
e) Check to make sure C is perpendicular to both A and B. (Hint: Use the dot
product! Between your new vector and A & B)
Page 7 of 9
Name ___________________________________
7. (20 points) A particle starts from the origin at t = 0 s and moves along the positive xaxis. A graph of the velocity of the particle as a function of time is shown below. The v-axis
scale is set by v_s = 28.0 m/s. This is the value at the top of the axis, not the value
of each box.
a) What is the velocity of the particle at t = 11 s? (Nothing to calculate. READ
the graph)
b) What is the acceleration of the particle at t = 11 s? (Slope?)
c) What is the position of the particle at t = 12 s? (i.e. it’s displacement from t =
0 s until t = 12 s) (integral?)
d) What is the average acceleration of the particle between t = 2 s and t = 12
s?
e) What is the average velocity of the particle between t = 2 s and t = 16 s?
Page 8 of 9
Name ___________________________________
Extra Credit (10 points) Navigating in the Big Dipper. All the stars of the Big Dipper (part of
the constellation Ursa Major) may appear to be the same distance from the earth, but in fact they
are very far from each other. Figure P1.101 shows the distances from the earth to each of these
stars. The distances are given in light-years (ly), the distance that light travels in one year. One
light-year equals 9.461 * 1015 m. (a) Alkaid and Merak are 29.6o apart in the earth’s sky. In a
diagram, show the relative positions of Alkaid, Merak, and our sun. Find the distance in light-years
from Alkaid to Merak. (b) To an inhabitant of a planet orbiting Alkaid, how many degrees apart in
the sky would Merak and our sun be?
Hint: Treat the Sun & Earth to be at the same point. The distance between them is negligible
compared to the distance between the stars.
Also: You can use trig (Law of Cosines?) or components in a coordinate system…(maybe put one
distance along an axis? 😉 )
SHOW ALL WORK
Page 9 of 9
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