lecture 3: free fall & vectors in physics (sections 2.6-2.7, 3.1-3.6 )
TRANSCRIPT
Lecture 3:Free Fall & Vectors in Physics
(sections 2.6-2.7, 3.1-3.6)
Freely Falling ObjectsFree fall from rest:
Free fall is the motion of an object subject only to the influence of gravity. The acceleration due to gravity is a constant, g.
g = 9.8 m/s2
For free falling objects, assuming your x axis is
pointing up, a = -g = -9.8 m/s2
Free-fall must exclude air resistance
An object falling in air is subject to air resistance (and therefore is not freely falling).
1-D motion of a vertical projectile
S
1-D motion of a vertical projectilev
ta:
v
tb:
v
tc:
v
td:Question 1:
1-D motion of a vertical projectilev
ta:
v
tb:
v
tc:
v
td:Question 1:
Basic equations
20 0
0
2 20 0
1 ; ,
2 ; ,
2 ; ,
x t x v t at t x
v t v at t v
v t v a x t x x v
Free Fall
0
Let's define:
ˆ+x as downward
x=0 is point from which object dropped
t=0 is time object dropped
x t x 0v 21
2t gt
Freely falling Object - moreFreely falling Object - more
Freely falling Object – even moreFreely falling Object – even more
2
1 1 1
2
2 2 1 1
2
3 3 1 1
2
4 4 1 1
1 21
2
1 42 2 1.414
2
1 63 3 1.732
2
1 84 2 2.000
2
xx a t t t
a
xx a t t t t
a
xx a t t t t
a
xx a t t t t
a
Question 2 Free Fall I
a) its acceleration is constant everywhere
b) at the top of its trajectory
c) halfway to the top of its trajectory
d) just after it leaves your hand
e) just before it returns to your hand on the way down
You throw a ball straight up into the air. After it leaves your hand, at what point in its flight does it have the maximum value of acceleration?
The ball is in free fall once it is released. Therefore, it is entirely
under the influence of gravity, and the only acceleration it
experiences is g, which is constant at all points.
Question 2 Free Fall I
a) its acceleration is constant everywhere
b) at the top of its trajectory
c) halfway to the top of its trajectory
d) just after it leaves your hand
e) just before it returns to your hand on the way down
You throw a ball straight up into the air. After it leaves your hand, at what point in its flight does it have the maximum value of acceleration?
Question 3 Free Fall II
Alice and Bill are at the top of a building. Alice throws her ball downward. Bill simply drops his ball. Which ball has the greater acceleration just after release?
a) Alice’s ball a) Alice’s ball
b) it depends on how hard b) it depends on how hard the ball was thrownthe ball was thrown
c) neither—they both have c) neither—they both have the same accelerationthe same acceleration
d) Bill’s balld) Bill’s ball
v0
BillAlice
vA vB
Both balls are in free fall once they are
released, therefore they both feel the
acceleration due to gravity (g). This
acceleration is independent of the initial
velocity of the ball.
Alice and Bill are at the top of a building. Alice throws her ball downward. Bill simply drops his ball. Which ball has the greater acceleration just after release?
a) Alice’s ball
b) it depends on how hard the ball was thrown
c) neither—they both have the same acceleration
d) Bill’s ball
v0
BillAlice
vA vB
Follow-up: which one has the greater velocity when they hit the ground?
Question 3 Free Fall II
You drop a rock off a bridge. When the rock has fallen 4 m, you drop a second rock. As the two rocks continue to fall, what happens to their separation?
a) the separation increases as they fall
b) the separation stays constant at 4 m
c) the separation decreases as they fall
d) it is impossible to answer without more information
Question 4 Throwing Rocks I
At any given time, the first rock always has a greater velocity than the second rock, therefore it will always be increasing its lead as it falls. Thus, the separation will increase.
You drop a rock off a bridge. When the rock has fallen 4 m, you drop a second rock. As the two rocks continue to fall, what happens to their separation?
a) the separation increases as they fall
b) the separation stays constant at 4 m
c) the separation decreases as they fall
d) it is impossible to answer without more information
Question 4 Throwing Rocks I
A hot-air balloon has just lifted off and is rising at the constant rate of 2.0 m/s. Suddenly one of the passengers realizes she has left her camera on the ground. A friend picks it up and tosses it straight upward with an initial speed of 13 m/s. If the passenger is 2.5 m above her friend when the camera is tossed, how high is she when the camera reaches her?
Solution: we know how to get position as function of time
balloon
camera
Find the time when these are
equal
0.26 or 2.0t s s
Scalars Versus VectorsScalar: number with units
Example: Mass, temperature, kinetic energy
Vector: quantity with magnitude and direction
Example: displacement, velocity, acceleration
Vector addition
A
B
C
C = A + B
C = A + Btail-to-head visualization
Parallelogram visualization
Adding and Subtracting Vectors
BA
Adding and Subtracting Vectors
D = A - BIf
then D = A +(- B)
C = A + B
D = A - B
-B is equal and opposite to B
If two vectors are given
such that A + B = 0,
what can you say about
the magnitude and
direction of vectors A and
B?
a) same magnitude, but can be in any
direction
b) same magnitude, but must be in the same direction
c) different magnitudes, but must be in the same direction
d) same magnitude, but must be in opposite directions
e) different magnitudes, but must be in opposite directions
Question 5Question 5 Vectors IVectors I
If two vectors are given
such that A + B = 0,
what can you say about
the magnitude and
direction of vectors A and
B?
a) same magnitude, but can be in any
direction
b) same magnitude, but must be in the same direction
c) different magnitudes, but must be in the same direction
d) same magnitude, but must be in opposite directions
e) different magnitudes, but must be in opposite directions
The magnitudes must be the same, but one vector must be pointing
in the opposite direction of the other in order for the sum to come
out to zero. You can prove this with the tip-to-tail method.
Question 5Question 5 Vectors IVectors I
The Components of a VectorCan resolve vector into perpendicular components using a two-dimensional coordinate system:
characterize a vector using magnitude |r| and direction θr
or by using perpendicular components rx and ry
Calculating vector componentsLength, angle, and components can be calculated from each other using trigonometry:
A2 = Ax2 + Ay
2
Ax = A cos θ
Ay = A sin θ
tanθ = Ay / Ax
Ax
Ay
Magnitude (length) of a vector A is |A|, or simply A
relationship of magnitudes of a vector and its component
The Components of a Vector
Signs of vector components:
Adding and Subtracting Vectors1. Find the components of each vector to be added.2. Add the x- and y-components separately.3. Find the resultant vector.
Scalar multiplication of a vector
Multiplying unit vectors by scalars: the multiplier changes the length, and the sign indicates the direction.
Unit Vectors
Unit vectors are dimensionless vectors of unit length.
A
Ax = Ax x^
Ay = Ay y
Question 6Question 6 Vector AdditionVector Addition
You are adding vectors of length 20 and 40 units. Of the following choices, only one is a possible result for the magnitude. Which is it?
a) 0a) 0
b) 18b) 18
c) 37c) 37
d) 64d) 64
e) 100e) 100
Question 6Question 6 Vector AdditionVector Addition
a) 0a) 0
b) 18b) 18
c) 37c) 37
d) 64d) 64
e) 100e) 100
The minimumminimum resultant occurs when the vectors
are oppositeopposite, giving 20 units20 units. The maximummaximum
resultant occurs when the vectors are alignedaligned,
giving 60 units60 units. Anything in between is also
possible for angles between 0° and 180°.
You are adding vectors of length 20 and 40 units. Of the following choices, only one is a possible result for the magnitude. Which is it?
Displacement and change in displacement
Position vector points from the origin to a location.
The displacement vector points from the original position to the final position.
Average Velocity
t1
t2
Average velocity vector:
So is in the same
direction as .
Instantaneous
velocity vector v
is always tangent
to the path.
Instantaneous
t1
t2
Average Acceleration
Average acceleration vector is in the direction of the change in velocity:
Instantaneous acceleration
Velocity vector is always in the direction of motion; acceleration vector can points in the direction velocity is changing:
Velocity and Acceleration
Question 6:Only one vector shown here can represent accelerationif the speed is constant.Which is it?
a) 1
b) 2
c) 3
d) 4
Relative Motion
The speed of the passenger with respect to the ground depends on the relative directions of the passenger’s and train’s speeds:
Velocity vectors can add, just like displacement vectors
Relative Motion
This also works in two dimensions:
You are riding on a Jet Ski at an angle of 35° upstream on a river flowing with a speed of 2.8 m/s. If your velocity relative to the ground is 9.5 m/s at an angle of 20.0° upstream, what is the speed of the Jet Ski relative to the water? (Note: Angles are measured relative to the x axis shown.)
Now suppose the Jet Ski is moving at a speed of 12 m/s relative to the water. (a) At what angle must you point the Jet Ski if your velocity relative to the ground is to be perpendicular to the shore of the river? (b) If you increase the speed of the Jet Ski relative to the water, does the angle in part (a) increase, decrease, or stay the same? Explain. (Note: Angles are measured relative to the x axis shown.)
- Assignment 2 on MasteringPhysics. Due Monday, September 6.
- Reading, for next class (4.1-4.5)
- When you exit, please use the REAR doors!