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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