kˆjˆמ zyxr ++= о

kˆ jˆ

iˆ y z

x

r r = + +

1

2 r

r

rr r r

−

=

∆

Position

Displacmen

kˆ jˆ

iˆ and

kˆ jˆ

iˆ _{1 1 2 2 2 2}

1

1 x y z r x y z

rr = + + r = + +

) kˆ jˆ

iˆ ( - ) kˆ jˆ

iˆ

(x₂ y₂ z₂ x₁ y₁ z₁

r = + + + +

∆r

kˆ ) (

jˆ ) - (

iˆ ) -

(x₂ x₁ y₂ y₁ z₂ z₁

r = + + −

∆r

1 2

kˆ jˆ

iˆ y z

x

r = ∆ + ∆ + ∆

∆r

Average velocity Instantaneous velocity (Or velocity)

but

Example:

The direction of the Instantaneous velocity

Average Acceleration Instantaneous Acceleration (Or acceleration)

but

y

Max. height Projectile

x θ0

Projectile’s Path

Launching point Landing point

Projectile

R

The range R

y

x θ0

v_0x v_0y

y

x x

y y

y

x x

y

v v

v_y

θ0

x v₀

v

v_0x v_0y

v_x v_x

v_y

y

Projectile motion

Horizontal Motion Vertical Motion

v_x

v_y v_y

Projectile motion

Horizontal Motion Vertical Motion

v=

v

v_x v_x

Max. height

v_y

V r

V r

θ0

v_0x v_0y

x x

y y

x

V0

r V

y

H

Max. height

x

x

R R

x

The equation of the projectile path (TRAJECTORY)

This is the equation of a parabola, so the projectile path is parabolic

x

R R

x

1-Velocity :

-magnitude constant v.

-direction :tangent to the circle in the direction of motion.

V r

V r

ar

ar ar

A particle is in uniform circular motion if it travels around a circle or circular arc at constant speed.

direction of motion.

V V r

r

2- Acceleration:

Why is the particle accelerating even though the speed does not vary?

- magnitude

r a v

2

=

- It is called Centripetal acceleration(meaning seeking center) - direction: toward the center.

ar

3- Period: is the time for a particle go around the circle once.

velocity distance Time =

v T ₂π r

=

For one round ⇒ distance = circumference of the circle

j v i

v

vr = _xˆ + _y ˆ

jˆ v vr = iˆ

v vr = −

jˆ v vr = −

iˆ v vr =

j a i

a

ar = _xˆ + _y ˆ

iˆ a ar =

iˆ a ar = − jˆ

a ar = −

jˆ a ar =