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Welcome back to Physics 211. Today’s agenda: Velocity and acceleration in two-dimensional motion Motion under gravity Accel. on curved path. Reminder. Homework due this week:

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Welcome back to physics 211

Welcome back to Physics 211

Today’s agenda:

Velocity and acceleration in two-dimensional motion

Motion under gravity

Accel. on curved path


Reminder

Homework due this week:

  • Wed: Tutorial Homework on Acceleration in one dim. (questions 1-5)(p. 13 - 16 in HW volume)


Homework grading
Homework grading

  • Parts of Tutorial home-works are graded in detail. 0 - 3 points given based on correctness and completeness.


Exam 1 next thursday 09 22 05
Exam 1: next Thursday (09/22/05)

  • Seating arrangement will be announced.

  • Material covered:

    • Textbook chapters 1, 2, and 3

    • Lectures up to 09/20 (slides online)

    • Tutorials on Velocity, Acceleration in one dimension, and Motion in two dimensions

    • Problem Solving Activities 1, 2, and 3 (on Graphs,Vectors, and Problems on motion in two dimensions)

    • Tutorial homeworks

    • Worksheet on Kinematics-in-Context(next W’day)


Describing motion with vectors

  • Positions and displacements

    Ds = sF - sI

  • Velocities and changes in velocity:

    vav= Ds/Dt vinst= limDt->0Ds/Dt

    Dv = vF - vI

  • aav= Dv/Dt ainst= limDt->0Dv/Dt


2d motion
2D Motion

Note:

component of position vector

along x direction is the x coordinate!

y

s – vector position

s=xi+yj  v=vxi+vyj

x


2d motion in components
2D Motion in components

  • x and y motions decouple

    • vx=Dx/Dt; vy=Dy/Dt

    • ax=Dvx/Dt; ay=Dvy/Dt

  • If acceleration is only non-zero in 1 direction – can choose coordinates so that 1 component of accel. is zero

    • Eg . motion under gravity


Simplest case
Simplest case

  • 2D motion with constant acceleration

    • Describes motion of ball under gravity (close to surface of Earth)

    • i.e x and y components of position vector satisfy const accel. equations …


Motion under gravity
Motion under gravity

ax=0

vFx=vIx

xF=xI+vIxt

ay=-g

vFy=vIy-gt

yF=yI+vIyt-1/2gt2

y

vIy=vsin(q)

vIx=vcos(q)

v

q

x


Ball A is released from rest. Another identical ball (ball B) is thrown horizontally at the same time and from the same height. Which ball will reach the ground first?

1. Ball A

2. Ball B

3. Both balls reach the ground at the same time.

4. The answer depends on the initial speed of ball B.


Demo 2 balls time to hit ground indep of v x
DEMO: 2 balls – time to hit ground indep of v B) is thrown horizontally at the same time and from the same height. Which ball will reach the ground first?x


A ball is thrown vertically upward from a cart at rest. The ball goes up, reaches its highest point and returns to the cart.

In a second experiment, the cart is moving at constant velocity and the ball is thrown in the same way, where will the ball land?

1. In front of the cart.

2. Behind the cart.

3. Inside the cart.

4. The outcome depends on the speed of the cart.


Demo projecting a ball from a moving cart
DEMO: projecting a ball from a moving cart … ball goes up, reaches its highest point and returns to the cart.


Projectile motion
Projectile motion ball goes up, reaches its highest point and returns to the cart.

R : when is y=0 ?

t(vy1-1/2gt)=0

=> hmax (y eqn)

i.e T=2vsinq/g

=> R (x eqn.)


Maximum height and range
Maximum height and range ball goes up, reaches its highest point and returns to the cart.


A battleship simultaneously fires two shells at enemy ships. If the shells follow the parabolic trajectories shown, which ship will be hit first?

1. A

2. Both at the same time

3. B

4. need more information


Motion on a curved path at constant speed
Motion on a curved path If the shells follow the parabolic trajectories shown, which ship will be hit first?at constant speed

Is the acceleration of the object equal to zero?


Velocity is tangent to path
Velocity is tangent to path If the shells follow the parabolic trajectories shown, which ship will be hit first?

Ds

sI

sF

O

v = Ds/Dt lies along

dotted line. As Dt0

direction of v is tangent

to path


Motion on a curved path at constant speed1
Motion on a curved path If the shells follow the parabolic trajectories shown, which ship will be hit first?at constant speed


Another way to add vectors
Another way to add vectors If the shells follow the parabolic trajectories shown, which ship will be hit first?

vF+(-vI)=Dv

Dv

vI

vF

same as

-vI

vF

Dv



For which of the following motions of a car does the change in velocity vector have the greatest magnitude? (All motions occur at the same constant speed.)

1. A 90° right turn at constant speed

2. A U-turn at constant speed

3. A 270° turn on a highway on-ramp

4. The change in velocity is zero for all three motions.


1. A 90° right turn at constant speed in velocity vector have the greatest magnitude? (All motions occur at the same constant speed.)

2. A U-turn at constant speed

3. A 270° turn on a highway on-ramp


A car moves along the path shown. Velocity vectors at two different points are sketched. Which of the arrows below most closely represents the direction of the average acceleration?

1. 2.

3. 4.


A child is riding a bicycle on a level street. The velocity and acceleration vectors of the child at a given time are shown.

Which of the following velocity vectors may represent the velocity at a later time?

a

1. 2.

3. 4.


Velocity of child on bicycle
Velocity of child on bicycle and acceleration vectors of the child at a given time are shown.

a


The initial velocity of a bird is 16 m/s to the West. For 4 seconds, it experiences an acceleration of 2 m/s2 to the North. (As on most maps, North is represented by an arrow pointing towards the top of the page.)

Which of the vectors below best describes the bird’s final velocity?

1. 3.

2. 4.


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