Projectile Motion

Major Principles for All Circumstances•Horizontal motion is constant velocity

•Vertical motion is constant downward acceleration

•a = g = -9.8 m/s2

Projectile Motion

The Big Four + One MoreIn x

•x = vxt

In y

•y = voyt + 1/2 gt2

•vy = voy + gt

•vy2 = voy

2 + 2gy

•y = 1/2(voy + vy)t

Ties them togetherTime & Launch Angle

Projectile Motion

Types of Projectile Problems

•Type A - Half of a Parabola

•Type B -

•Type C -

Full Parabola - Symmetric

Partial or Asymmetric Parabola

Type A - Half of a ParabolaProjectile Motion

In the vertical direction

•The object acts like a dropped object

•Initial vertical velocity is zero; voy = 0

Type A - Half of a ParabolaProjectile Motion

To solve for time, often you will use...

y = voyt + 1/2 gt2

since voy = 0

y = 1/2 gt2 t = (2y/g)

Therefore...

WARNING:

Be careful using shortcut formulas!!!!

Type A - Half of a ParabolaProjectile Motion

If the problem is reversed...Romeo throws a rock up to Juliet; hits window horizontally

Because of symmetry, just solve the problem backwards, make voy = 0

Projectile MotionType B - Full Parabola

Notice the ball lands back in the truck...

only if the truck moves with constant velocity

Projectile MotionType B - Full Parabola

If you solve for the full parabola...

The vertical displacement is zero; y = 0

The time is the total hang time

Projectile MotionType B - Full Parabola

If you solve for half the parabola...

The vertical velocity at the peak is; vy = 0

The time is equal to half the hang time

Projectile MotionType B - Full Parabola

The Range Formula

vo

WARNING:

Use the triangle for velocities only!!!!

voy = vosinvx = vocos

Projectile MotionType B - Full Parabola

The Range Formula

vx = vocos

x = vxtx = (vocos)t

voy = vosin

Projectile MotionType B - Full Parabola

The Range Formulax = vxt

voy = vosin

x = (vocos)t

vy = voy + gt-voy = voy + gt-2voy = gt-(2voy)/g = t

vy = -voy

-(2vosin)/g = t

Projectile MotionType B - Full Parabola

The Range Formulax = vxtx = (vocos)t

-(2vosin)/g = t

x = (vocos)(-2vosin/g)

Projectile MotionType B - Full Parabola

The Range Formulax = vxtx = (vocos)t

x = (vocos)(-2vosin/g)

Projectile MotionType B - Full Parabola

The Range Formulax = vxtx = (vocos)t

x = (vocos)(-2vosin/g)

x = -vo2(2sincos)/g

Trig Identity: 2sincos = sin2

x = -vo2sin2/g

Projectile MotionType B - Full Parabola

The Range Formulax = vxtx = (vocos)t

x = (vocos)(-2vosin/g)

x = -vo2(2sincos)/g

x = -vo2sin2/g

Projectile MotionType B - Full Parabola

The Range Formulax = vxtx = (vocos)t

x = (vocos)(-2vosin/g)

x = -vo2(2sincos)/g

x = -vo2sin2/g

WARNING:

Be careful using shortcut formulas!!!!

•Optimum Angle of 45

Maximum range

Projectile MotionType B - Full Parabola

•Supplementary Angles

Equal ranges

Projectile MotionType C - Partial or Asymmetric Parabola

•Each problem is unique, so take your time and...

•Some problems can be treated as two Type A problems

stick to your major principles from the beginning

Projectile MotionType C - Partial or Asymmetric Parabola

We don’t know time, but we must find out the height (y) of an object.

Very Unique Equation

y = voyt + 1/2gt2

Projectile MotionType C - Partial or Asymmetric Parabola

Very Unique Equation

y = voyt + 1/2gt2

vo

voy = vosinvx = vocos

y = (vosin)t + 1/2gt2

Projectile MotionType C - Partial or Asymmetric Parabola

Very Unique Equation

y = voyt + 1/2gt2

vo

voy = vosinvx = vocos

y = (vosin)t + 1/2gt2

x = vxtx = (vocos)tx/(vocos) = t

Projectile MotionType C - Partial or Asymmetric Parabola

Very Unique Equation

y = voyt + 1/2gt2

y = (vosin)t + 1/2gt2

x/(vocos) = t

y = vosin(x/(vocos)) + 1/2g(x/(vocos))2

Projectile MotionType C - Partial or Asymmetric Parabola

Very Unique Equation

y = voyt + 1/2gt2

y = (vosin)t + 1/2gt2

y = vosin(x/(vocos)) + 1/2g(x/(vocos))2

y = x(sin/cos) + 1/2g(x2/(vo2cos2))

y = xtan + gx2/(2vo2cos2)

Projectile MotionType C - Partial or Asymmetric Parabola

Very Unique Equation

y = voyt + 1/2gt2

y = (vosin)t + 1/2gt2

y = vosin(x/(vocos)) + 1/2g(x/(vocos))2

y = x(sin/cos) + 1/2g(x2/(vo2cos2))

y = xtan + gx2/(2vo2cos2)

Works for all types of

problems, !!

Most useful with Type C!!