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Data Accurate Measurements

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.”

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Data Accurate Measurements

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  1. DataAccurate Measurements

  2. The Protractor • Tycho Brahe 1546 –1601 • Accuracy to one minute of arc • 1/60 degree • = 1 nautical mile • sextant

  3. 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

  4. 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

  5. Formulas • ∆ v = g ∆t change in v ~ time • v2 − v1 = g (t2 − t1) (time when) • v2 = v1 + gt (elapsed time) • vavg= v̄ = ½ (v1 + v2) • distance = s = ∆x = x2 − x1 = v̄ t • s= ½ (v1 + v2) t • s = ½ (v1 + v1 + gt) t • s = v1 t + ½ gt2

  6. 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

  7. 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 = v̅ t = ½ (40+10) ⋅t = 25m/s ⋅ 3 s = 75 m

  8. 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

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