Motion in 1D:Graphing {x,v,a} vs t
𝑣=

𝑑𝑥
𝑑𝑡
lim 𝑥
=
= Slope of x-t graph
t0 𝑡
a=
𝑑𝑣
𝑑𝑡
lim 𝑣
=
= Slope of v-t graph
t0 𝑡
average vs. instantaneous value of
velocity, acceleration
"acceleration is not velocity, …."
Kinematic Equations

Used in situations with uniform
acceleration
v  vo  at
1
x  vt  vo  v  t
2
1 2
x  vot  at
2
v 2  vo2  2ax
Adding Vectors
• Graphical / geometrical
• Algebraic : by adding components
• Unit vectors
A = Ax 𝑖 +Ay 𝑗 +Az𝑘
• Algebra, trigonometry
(sin,cos,tan,pythagoras)
Motion in 2D


x,v,a  𝑟, 𝑣, 𝑎
𝑟 = x 𝑖 + y𝑗 + z𝑘
x  𝑟
𝑟, 𝑣 , 𝑎
𝑣 points along tangent to path
All definitions in 1D  2D
Motion in 2D
• Projectile motion
• Treat x- and y- motion separately
• ax = 0, |ay| = |g|
• Uniform circular motion
• a = v2/r, points towards center of circle
An object's acceleration vs. time is:
Which graph best represents the object's
velocity vs. time?
At t = 0, a projectile is fired straight upward from a cannon.
It goes up and then comes down. Assume that there is no air
resistance. Upward is chosen as the positive y-direction.
Which of the following graphs most accurately shows
velocity vs. time for the projectile during its flight?
The vector A has length 5. The vector B has magnitude 4. You
have no other information about the two vectors. Which one of
the following statements about the vector A+B is true?
A) The magnitude of the vector A+B must be between 5 and 9.
B) The magnitude of the vector A+B must be between 0 and 4.
C) The magnitude of the vector A+B must be between -1 and 9.
D) The magnitude of the vector A+B must be between 1 and 9.
Suppose you know the velocity vector of a particle at a particular
instant of time and it is parallel to the ground. What do you know
about the acceleration a of the particle?
A) The velocity vector and the acceleration vector are always
either in the same direction or in opposite directions.
B) The acceleration vector is straight down.
C) You know nothing at all about the acceleration vector.
D) The acceleration must be zero.
The x- and y-coordinates of a particle as a function of time are
x(t )  b  c t, y(t )  d  e t , where b,c,d, and e are positive constants.
Which arrow could be the velocity of the particle?
y
B
A
E: (it's zero)
C
D
x
b) Which direction is the acceleration of the particle?
A flare is dropped from an airplane flying at
uniform velocity (constant speed in a straight
line). Neglecting air resistance, the flare will
A: quickly lag behind the plane
B: remain vertically under the plane
C: move ahead of the plane
D: it depends how fast the plane is flying.
A race car is driving around a circular race-track
with a constant speed of 100 mph. The car
A: is accelerating
B: is not accelerating.
C: (not enough information given)
An elevator is going up at a constant speed. Near
the top floor, it starts to slow to a stop. During
the period that it is slowing down, its
acceleration is
•
A:
B:
C:
D:
downward
upward.
in some other direction.
impossible to tell without more information
Two stones are dropped into a bottomless pit,
the second stone is dropped 2 seconds after the
first stone. Assume no air resistance. As both
stones fall, the difference in their velocities..
A: increases
B: decreases
C: remains constant