Lecture 3:

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

1-D motion of a vertical projectile

2 20 0 0 0

1 1;

2 2x t x v t at x t t gt ax v g

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

A ball is dropped from a height of 5.0 m.1.How long does it take to reach the floor?2.How fast will it be going when it hits?3.At what height will it be going half this speed?

A ball is dropped from a height of 5.0 m.1.How long does it take to reach the floor? {t,y}2.How fast will it be going when it hits the floor? {v,t}3.At what height will it be going half this speed? {y,v}

0 0

2

0

2 2 220 1 1 1

2 2 2

, 0,

1. ,

1 2 2 7.00 1.2

2 9.8

2. ,

9.8 1.2 12 12

3. ,

1 1 127.0 3.5

2 2 9.8 2

f f f

f f f f f

f f

y h v a g

y t

hy t h gt t s

g

v t

v t v v gt gt m s v m s

v y

v vv g y h g h y y h m

g

A ball is dropped from a height of 7.0 m.1.How long does it take to reach the floor?2.How fast will it be going when it hits?3.At what height will it be going half this speed?

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?

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

Recall: 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

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 VectorsUnit 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 position

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:

2-Dimensional Motion

(sections 4.1-4.5)

A certain vector has A certain vector has xx and and yy components components

that are equal in magnitude. Which of the that are equal in magnitude. Which of the

following is a possible angle for this vector following is a possible angle for this vector

in a standard in a standard x-yx-y coordinate system? coordinate system?

a) 30°

b) 180°

c) 90°

d) 60°

e) 45°

Question 7: Question 7: Vector Components IIVector Components II

A certain vector has A certain vector has xx and and yy components components

that are equal in magnitude. Which of the that are equal in magnitude. Which of the

following is a possible angle for this vector following is a possible angle for this vector

in a standard in a standard x-yx-y coordinate system? coordinate system?

a) 30°

b) 180°

c) 90°

d) 60°

e) 45°

The angle of the vector is given by tan Θ = y/x. Thus, tan Θ =

1 in this case if x and y are equal, which means that the angle

must be 45°.

Question 7: Question 7: Vector Components IIVector Components II

a) point 1

b) point 2

c) point 3

d) point 4

e) I cannot tell from that graph.

Question 8: Acceleration and Velocity VectorsQuestion 8: Acceleration and Velocity VectorsBelow is plotted the trajectory of a particle in two dimensions, along with instantaneous velocity and acceleration vectors at 4 points. For which point is the particle speeding up?

a) point 1

b) point 2

c) point 3

d) point 4

e) I cannot tell from that graph.

Question 8: Acceleration and Velocity VectorsQuestion 8: Acceleration and Velocity VectorsBelow is plotted the trajectory of a particle in two dimensions, along with instantaneous velocity and acceleration vectors at 4 points. For which point is the particle speeding up?

At point 4, the acceleration and velocity point in the same direction, so the particle is speeding up

The Components of Velocity Vector

Motion along each direction becomes a 1-D problem

vx

vyv

Projectile Motion: objects moving under gravity

Assumptions:

• ignore air resistance

• g = 9.81 m/s2, downward

• ignore Earth’s rotation

• y-axis points upward, x-axis points horizontally

• acceleration in x-direction is zero

• Acceleration in y-direction is -9.81 m/s2

x

y g

vy

vx

These, then, are the basic equations of projectile motion:

Launch angle: direction of initial velocity with respect to horizontal

Zero Launch Angle

In this case, the initial velocity in the y-direction is zero. Here are the equations of motion, with x0 = 0 and y0 = h:

Zero Launch Angle

Eliminating t and solving for y as a function of x:

This has the form y = a + bx2, which is the equation of a parabola.

The landing point can be found by setting y = 0 and solving for x:

Trajectory of a zero launch-angle projectile

horizontal points equally spaced

vertical points not equally spaced

parabolic

y = a + bx2

ball 1

ball 2

Where will Ball 1 land on the lower surface?

a) Ahead (to the left) of Ball 2

b) Behind (to the right) of Ball 2

c) On top of Ball 2

d) impossible to say from the given information

Question 9: Drop and not

ball 1

ball 2

Where will Ball 1 land on the lower surface?

a) Ahead (to the left) of Ball 2

b) Behind (to the right) of Ball 2

c) On top of Ball 2

d) impossible to say from the given information

Question 9: Drop and not

x

y

vx

vx

vy=0

vy=0

x

y

vy=0

vy=0

Δx

vx

vx

vyv

x

vy=0

vy=0

v

A ball is projected horizontally at the same time as one is dropped from the same height. Which will hit the floor first?

A ball is projected horizontally at the same time as one is dropped from the same height. Which will hit the floor first?

0

0

0

0

0 0

0

2

2

Ball 1:

0

0

0 0

0

10

21

02

2

x

x

y

y

f f

f

x

v v

a

x t v t v t

y h

v

a g

y t h gt

y t h gt

ht

g

0

0

0

0

2

2

Ball 2:

0

0

0

0 0 0 0

0

10

21

02

2

x

x

y

y

f f

f

x

v

a

x t

y h

v

a g

y t h gt

y t h gt

ht

g

-g

v0Sin(θ)

v0Cos(θ)

General Launch Angle

In general,

v0x = v0 cos θ and

v0y = v0 sin θ

This gives the equations of motion:

Range: the horizontal distance a projectile travels

As before, use

and

Eliminate t and solve for x when y=0

(y = 0 at landing)

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.)

Question 11: Relativity Car

A small cart is rolling at

constant velocity on a flat track. It fires a ball straight up into the air as it moves. After it is fired, what happens to the ball?

a) it depends on how fast the cart is moving

b) it falls behind the cart

c) it falls in front of the cart

d) it falls right back into the cart

e) it remains at rest

A small cart is rolling at

constant velocity on a flat track. It fires a ball straight up into the air as it moves. After it is fired, what happens to the ball?

a) it depends on how fast the cart is moving

b) it falls behind the cart

c) it falls in front of the cart

d) it falls right back into the cart

e) it remains at rest

when viewed from train

when viewed from ground

In the frame of reference of the cart, the ball only has a vertical component of velocity. So it goes up and comes back down. To a ground observer, both the cart and the ball have the same horizontal velocity, so the ball still returns into the cart.

Question 11: Relativity Car

- Assignment 1 on MasteringPhysics due Friday, September 6 (12:59 pm).

- Exit using the rear doors!