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PPT – 3. Motion in Two and Three Dimensions PowerPoint presentation | free to download - id: 5b153d-NjcyZ

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3. Motion in Two and Three Dimensions

Recap Constant Acceleration

Area under the function v(t).

Recap Constant Acceleration

Recap Acceleration due to Gravity (Free Fall)

- In the absence of air resistance all objects fall

with the same constant acceleration of about

g 9.8 m/s2 - near the Earths surface.

Recap Example

A ball is thrown upwards at 5 m/s, relative to

the ground, from a height of 2 m.

We need to choose a coordinate system.

Recap Example

Lets measure time from when the ball is

launched. This defines t 0.

Lets choose y 0 to be ground level and up to

be the positive y direction.

Recap Example

1. How high above the ground will the ball

reach?

with a g and v 0.

use

Recap Example

2. How long does it take the ball to reach

the ground?

Use

with a g and y 0.

Recap Example

3. At what speed does the ball hit the

ground?

Use

with a g and y 0.

Vectors

Vectors

A vector is a mathematical quantity that has two

properties direction and magnitude.

One way to represent a vector is as an arrow the

arrow gives the direction and its length

the magnitude.

Position

A position p is a vector its direction is from o

to p and its length is the distance from o to p.

A vector is usually represented by

a symbol like .

Displacement

Vector Addition

The order in which the vectors are added does

not matter, that is, vector addition is

commutative.

Vector Scalar Multiplication

a and q are scalars (numbers).

Vector Subtraction

If we multiply a vector by 1 we reverse

its direction, but keep its magnitude the same.

Vector subtraction is really vector addition

with one vector reversed.

Vector Components

Acos? is the component, or the projection,

of the vector A along the vector B.

Vector Components

Vector Addition using Components

Unit Vectors

From the components, Ax, Ay, and Az, of a

vector, we can compute its length, A, using

If the vector A is multiplied by the scalar 1/A

we get a new vector of unit length in the same

direction as vector A that is, we get a unit

vector.

Unit Vectors

It is convenient to define unit vectors parallel

to the x, y and z axes, respectively.

Then, we can write a vector A as follows

Velocity and Acceleration Vectors

Velocity

Acceleration

Relative Motion

Relative Motion

Velocity of plane relative to air

Velocity of air relative to ground

Velocity of plane relative to ground

Example Relative Motion

A pilot wants to fly a plane due north Airspeed

200 km/h Windspeed 90 km/h direction W

to E 1. Flight heading? 2. Groundspeed?

Coordinate system î points from west to east

and j points from south to north.

Example Relative Motion

Example Relative Motion

Example Relative Motion

- Equate x components
- 0 200 sin ? 90
- ? sin-1(90/200)
- 26.7o west of north.

Example Relative Motion

Equate y components v 200 cos ? 179 km/h

Projectile Motion

Projectile Motion under Constant Acceleration

Coordinate system î points to the right, j

points upwards

Projectile Motion under Constant Acceleration

Impact point

R Range

Projectile Motion under Constant Acceleration

Strategy split motion into x and y components.

R Range R x - x0 h y - y0

Projectile Motion under Constant Acceleration

Find time of flight by solving y equation

And find range from

Projectile Motion under Constant Acceleration

Special case y y0, i.e., h 0

R

y(t)

y0

How to Shoot a Monkey

x 50 m h 10 m H 12 m Compute minimum initia

l velocity

H

Uniform Circular Motion

Uniform Circular Motion

O

r Radius

Uniform Circular Motion

Velocity

Uniform Circular Motion

d?/dt is called the angular velocity

Uniform Circular Motion

Acceleration

Assume d?/dt is constant

Uniform Circular Motion

Acceleration is towards center

Centripetal Acceleration

Uniform Circular Motion

The magnitudes of velocity and centripetal

acceleration are related as follows

Uniform Circular Motion

The magnitude of velocity and period T related

as follows

Summary

- In general, acceleration changes both the

magnitude and direction of the velocity. - Projectile motion results from the acceleration

due to gravity. - In uniform circular motion, the acceleration is

centripetal and has constant magnitude v2/r.

Uniform Circular MotionAlternative Derivation

Uniform Circular Motion

O

Now, take the derivative

r constant radius

and define the unit vector at right angles

to

P

Uniform Circular Motion

We can write

where R90 represents a 90o rotation

O

Since

it follows that

and so

P

where b is a constant

Uniform Circular Motion

The velocity can be written as

O

It is easy to show that b v/r

Now, take the derivative again to get the

acceleration

P

Uniform Circular Motion

The acceleration can be written as

O

The acceleration is centripetal

P

using the fact that R90R90 1

Uniform Circular Motion

The magnitudes of velocity and centripetal

acceleration are related as follows

O

P

Uniform Circular Motion

The magnitude of velocity and period T related

as follows

O

P