Data

Accurate Measurements

The Protractor

• Tycho Brahe 1546 –1601

• Accuracy to one minute of arc

• 1/60 degree• = 1 nautical mile• sextant

• Aristotle 384 – 322 BC• “Heavier things fall

faster”• Galileo Galilei 1564 –1642• “All objects fall with the

same constant acceleration.”

• The change in velocity is proportional to the elapsed time.

• Δv ~ Δt• Δv = g Δt∙

Acceleration

• Acceleration – How quickly the velocity changes

• a = Δv / t = (v2-v1)/t • Velocity – How quickly

the position changesv = s/t = Δx/t = (x2-x1)/t

• When things fall acceleration = a = 9.8 m/sec/sec ≈ 10 m /sec2

Formulas• ∆ v = g ∆t change in v ~ time• v2 − v1 = g (t2 − t1) (time when)

• v2 = v1 + gt (elapsed time)

• vavg = v� = ½ (v1 + v2)

• distance = s = ∆x = x2 − x1 = v� t

• s = ½ (v1 + v2) t

• s = ½ (v1 + v1 + gt) t

• s = v1 t + ½ gt2

More Formulas

• Algebra Review: (x + y ) (x – y ) f o i l∙

• = x² – xy + xy – y²• a = (v2 – v1)/t

• s = ½(v2+v1) t (average velocity) time ∙ ∙

• Multiply :• as = ½(v2² –v1²)

• 2as = v2² –v1²

Prob Set 4: 9

• An arrow is shot straight up with a speed of 40 meters per second. Three seconds later it hits a bird.

• How fast was the arrow going when it hit the bird ? • g = −10 m/s2

• t = 3 s• v0 = 40 m/s

• v1 = v0 + a⋅t = 40m/s – 10m/s2 ⋅ 3s =

• 40m/s – 30m/s = 10 m/s

• How high was the bird ?

• s = v% t = ½ (40+10) ⋅t = 25m/s ⋅ 3 s = 75 m

Suppose the Distance is Given

• The arrow is shot with a speed of 40 meters per second. The bird is 50 meters up.

• How fast was the arrow going when it hit the bird ? • g = −10 m/s2 s = 50 m t = ? • v1 = 40 m/s

• v2 = v1 + a ⋅ t Won’t work

• 2as = v2² – v1² v2 is unknown

• v2² = v1² + 2as = 40² + 2 ⋅ (–10) ⋅ 50

• = 1600 – 1000 = 600• v2 = √600 = 24.5 m/s up or 24.5 m/s down