Chapter 3: 2D Kinematics Thursday January 22nd

Reading: up to page 44 in the text book (Ch. 3)

• 1st Mini Exam (25 minutes) • Chapter 3: Motion in 2D and 3D

• Short Review • Review: Projectile motion • More example problems • Range of a projectile

• Uniform Circular Motion (if time) • Centripetal acceleration

Review: Components of vectors

cos

sin

x

y

a a

a a

θ

θ

=

=

Resolving vector components

The inverse process

2 2

tan

x y

y

x

a a a

aa

θ

= +

=

Review: Relative Motion

!vO'

Velocity of ball observed in O: !vb =!vb

' + !vO'

O’ x’

y’

O x

y

Two observers O and O’ ball

!vb

!vb'

Projectile motion • This series of photographic images illustrates the fact that vertical motion is unaffected by horizontal motion, i.e., the two balls accelerate downwards at the same constant rate, irrespective of their horizontal component of motion. • In all of the projectile motion problems that we will consider, we shall assume that the only acceleration is due to gravity (a=-g) which acts in the –y direction.

Equations of motion for constant acceleration

!r − !r0 =

!v0t + 12!at2

!v = !v0 +

!at

v2 = v0

2 + 2 !a ⋅(!r − !r0 )

(!r − !r0 ) = 1

2 (!v0 +!v )t

(!r − !r0 ) = !vt − 1

2!at2

(!r − !r0 )

!v

t

!a

!v0

Equation Missing quantity

Equation number

3.8

3.9

Important: equations apply ONLY if acceleration is constant.

Equations of motion for constant acceleration

0x x−

vx

t

ax

0v

Equation Missing quantity

Equation number

2.7

2.10

2.11

2.9

Important: equations apply ONLY if acceleration is constant.

These equations work the same in any direction, e.g., along x, y or z.

vx = v0x + axt

x − x0 = v0xt +12 axt

2

vx2 = v0x

2 + 2ax (x − x0 )

x − x0 =12 (v0x + vx )t

x − x0 = vxt −12 axt

2

Equations of motion for constant acceleration

210 0 2y yy y v t a t− = +

0y y yv v a t= +

2 20 02 ( )y y yv v a y y= + −1

0 02 ( )y yy y v v t− = +21

0 2y yy y v t a t− = −

0y y−

yv

t

ya

0 yv

Equation Missing quantity

Equation number

2.7

2.10

2.11

2.9

These equations work the same in any direction, e.g., along x, y or z.

Important: equations apply ONLY if acceleration is constant.

Equations of motion for constant acceleration

y − y0 = v0 yt − 1

2 gt2 vy = v0 y − gt

vy

2 = v0 y2 − 2g( y − y0 )1

0 02 ( )y yy y v v t− = +

y − y0 = vyt + 1

2 gt2

0y y−

yv

t

ya

0 yv

Equation Missing quantity

Equation number

Special case of free-fall under gravity, ay = -g. g = 9.81 m/s2 here at the surface of the earth.

Demonstration

Projectile motion

• Motion in a vertical plane where the only influence is the constant acceleration due to gravity. • In projectile motion, the horizontal motion and vertical motion are independent of each other, i.e. they do not affect each other.

• This feature allows us to break the motion into two separate one-dimensional problems: one for the horizontal motion; the other for the vertical motion. • We will assume that air resistance has no effect.

Analyzing the motion

0 0 0

0 0 0

cos

sin

x

y

v v

v v

θ

θ

=

=

Initial coordinates x0 and y0

Initial velocity = v0xi + v0 y j

Horizontal motion 0

0 0

0

cos

x x x

x

x

v v a t

a

v v θ

= +

= ⇒

=

x − x0 = v0xt +12 axt

2

x − x0 = v0 cosθ0( )t

Vertical motion

vy = v0 y + ayt

ay = −g ⇒

vy = v0 sinθ0 − gt

y − y0 = v0 yt + 12 ayt

2

y − y0 = v0 sinθ0( )t − 12 gt2

Maximum Range

The effects of air The physics professor's home run always goes further than the professional's

Uniform circular motion • Although the speed, v, does not change, the direction of the motion does, i.e., the velocity, which is a vector, does change.

• Thus, there is an acceleration associated with the motion.

• We call this a centripetal acceleration.

Same angles

Δr = rθ

Δv = vθ

⇒ Δvv

= Δrr

= vΔtr

Uniform circular motion • Although the speed, v, does not change, the direction of the motion does, i.e., the velocity, which is a vector, does change.

• Thus, there is an acceleration associated with the motion.

• We call this a centripetal acceleration.

ac =

v2

r(uniform circular motion)

Centripetal acceleration:

• A vector that is always directed towards the center of the circular motion, i.e., it’s direction changes constantly.

Uniform circular motion • Although the speed, v, does not change, the direction of the motion does, i.e., the velocity, which is a vector, does change.

• Thus, there is an acceleration associated with the motion.

• We call this a centripetal acceleration.

ac =

v2

r(uniform circular motion)

Centripetal acceleration:

Period: T = 2πr

v(sec) Frequency: f = 1

T= 1

2πvr

(sec−1)