Acceleration of free falling metal ball
Brian Ahn
November 11, 2013

Raw Data

<Table of time taken for ball to reach ground in different heights>

Height (m) ± 0.005 m | Time taken (s) ± 0.01 s | | Trial 1 | Trial 2 | Trial 3 | 0.5 | 0.35 | 0.35 | 0.33 | 1.0 | 0.45 | 0.43 | 0.40 | 1.5 | 0.53 | 0.44 | 0.47 | 2.0 | 0.56 | 0.56 | 0.53 | 2.5 | 0.72 | 0.64 | 0.72 | 3.0 | 0.75 | 0.76 | 0.79 | 3.5 | 0.77 | 0.84 | 0.78 |

Processing Data

<Table of average time taken for ball to reach ground in different heights>

Height (m) ± 0.005 | Average time taken (s) | 0.5 | 0.34 ± 0.01 | 1.0 | 0.43 ± 0.03 | 1.5 | 0.48 ± 0.05 | 2.0 | 0.55 ± 0.02 | 2.5 | 0.69 ± 0.04 | 3.0 | 0.77 ± 0.02 | 3.5 | 0.80 ± 0.04 |

Average of the time taken is (t1 + t2 + t3) ÷ 3

Uncertainty for the time taken is (tmax– tmin) ÷ 2

Average of the time taken in height 1 is (0.45 + 0.43 + 0.40) ÷ 3 = 0.42667 ≅ 0.43 (to two significant figures)

Uncertainty for the time taken in height 1 is (0.45 – 0.40) ÷ 2 = 0.025 ≅ 0.03 (to one significant figure)

<Table of average time taken squared by dropping an object in different heights>

Height (m) ± 0.005 | Average time taken squared (s2) | 0.5 | 0.12 ± 0.007 | 1.0 | 0.18 ± 0.03 | 1.5 | 0.23 ± 0.05 | 2.0 | 0.30 ± 0.02 | 2.5 | 0.48 ± 0.05 | 3.0 | 0.59 ± 0.03 | 3.5 | 0.64 ± 0.06 |

Uncertainty for average time taken squared is 2 × (percentage uncertainty) × (time taken) 2

Uncertainty for average time taken squared in height 0.5 = 2 × (0.01 ÷ 0.34) × 0.342 = 0.0068 ≅ 0.007 (to one significant figures)

Graphs

Graph of processed data

<Height versus average time taken squared for ball to reach ground with best fit drawn>

Gradient of the best linear fit = 0.19 (to two significant figures)

< Height versus average time taken squared for ball to reach ground with maximum gradient drawn >

Maximum gradient = 0.20

Maximum gradient = (0.7 – 0.113) ÷ (3.5 – 0.5) = 0.507 ÷ 3 = 0.20

< Height versus average time taken squared for ball to reach ground with minimum gradient drawn >

Minimum gradient = 0.15

Minimum gradient = (0.58 – 0.127) ÷ (3.5 – 0.5) = 0.453 ÷ 3 = 0.15

Uncertainty for gradient = (Gradientmax - Gradientmin) ÷ 2 = (0.20 – 0.15) ÷ 2 = 0.025 = 0.03 (to one significant figure)

Calculation

Formula of uniform acceleration with zero initial velocity

h = 12at2

Where h is height, a is acceleration, t is time

For graph height versus time squared, the gradient is t2h which is equal to 2a .

From the graph above, it is calculated that the gradient is 0.19 ± 0.03.

Therefore, 0.19 ± 0.03 = 2a .

a = 2 ÷ (0.19 ± 0.03)

Calculated acceleration = 20.19 = 11

Minimum acceleration = 20.22 = 9.1

Maximum acceleration = 20.16 = 13

Uncertainty for acceleration = (amax – amin) ÷ 2 = (13 – 9.1) ÷ 2 = 1.95 ≅ 2 (to one significant figure)

Acceleration is 11± 2

From the reference that universal gravitational acceleration is 9.8 ms-2, the acceleration calculated is quite accurate.

Bibliography

h = 12at2

universal gravitational acceleration is roughly 9.8 ms-2