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 = (v 2 -v 1 )/t Velocity – How quickly the position changes v = s/t = Δx/t = (x 2 -x 1 )/t When things fall acceleration = a = 9.8 m/sec/sec ≈ 10 m /sec 2

Formulas ∆ v = g ∆t change in v ~ time v 2 − v 1 = g (t 2 − t 1 ) (time when) v 2 = v 1 + gt (elapsed time) v avg = v̄ = ½ (v 1 + v 2 ) distance = s = ∆x = x 2 − x 1 = v̄ t s = ½ (v 1 + v 2 ) t s = ½ (v 1 + v 1 + gt) t s = v 1 t + ½ gt 2

More Formulas Algebra Review: (x + y )∙(x – y ) f o i l = x² – xy + xy – y² a = (v 2 – v 1 )/t s = ½(v 2 +v 1 )∙t (average velocity) ∙ time Multiply : as = ½(v 2 ² –v 1 ²)  2as = v 2 ² –v 1 ²

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/s 2 t = 3 s v 0 = 40 m/s v 1 = v 0 + a⋅t = 40m/s – 10m/s 2 ⋅ 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

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/s 2 s = 50 m t = ? v 1 = 40 m/s v 2 = v 1 + a ⋅ t Won’t work 2as = v 2 ² – v 1 ²  v 2 is unknown v 2 ² = v 1 ² + 2as = 40² + 2 ⋅ ( – 10) ⋅ 50 = 1600 – 1000 = 600 v 2 = √600 = 24.5 m/s up or 24.5 m/s down