Motion in One Dimension

Free Fall

dropping under gravity

A dropped object is constant acceleration with gravity doing the speeding up. With g = 10 the numbers stay clean.

mechanics Free fall reuses the constant-acceleration distance formula with gravity for the acceleration; g = 10 keeps it exact, and the gaps grow as it falls.

Example

A dropped object is just constant acceleration with gravity doing the speeding up. With g = 10 the numbers stay clean. Exact arithmetic here means exact results for the stated model inputs; measured inputs still carry uncertainty and significant-figure limits.

highlighted = computed this step

Set up the drop

A ball is dropped from rest and falls for 2 seconds. We use 10 metres per second squared for gravity to keep the arithmetic clean; the real value is about 9.8 metres per second squared, a little smaller.

g=10 m/s2,t=2 sg = 10\ \text{m}/\text{s}^{2}, \quad t = 2\ \text{s}
A ball at the moment it is droppedA ball held above the floor with a downward arrow showing the pull of gravity.ballg

Speed when it lands

Falling speed is gravity times time: 10 times 2 is 20 metres per second.

v=g t=10 m/s2 ⋅ 2 s=20 m/sv = g\,t = 10\ \text{m}/\text{s}^{2} \,\cdot\, 2\ \text{s} = \hl{20}\ \text{m}/\text{s}

Distance fallen

Free fall is the constant-acceleration distance with gravity for the acceleration: one half times 10 times the time squared, 4.

y=12 g t2=12 ⋅ 10 m/s2 ⋅ 4 my = \tfrac{1}{2}\,g\,t^{2} = \tfrac{1}{2} \,\cdot\, 10\ \text{m}/\text{s}^{2} \,\cdot\, 4\ \text{m}

Compute the fall

The ball falls 20 metres. As with any constant acceleration, equal times give growing gaps.

y=20 my = \hl{20}\ \text{m}
The ball falls in growing stepsA dropped ball shown at each whole second; the gaps grow and the downward velocity arrows lengthen as it speeds up toward the floor.t=0t=1ball

Fall distance grows faster than speed

Keep gravity fixed at 10 metres per second squared. Falling time makes speed grow evenly, but distance grows by the square of time.

tvy1 s10 m/s5 m2 s20 m/s20 m3 s30 m/s45 m\begin{array}{c|c|c}t & v & y \\ \hline 1\ \text{s} & 10\ \text{m}/\text{s} & 5\ \text{m} \\ 2\ \text{s} & 20\ \text{m}/\text{s} & 20\ \text{m} \\ 3\ \text{s} & 30\ \text{m}/\text{s} & 45\ \text{m}\end{array}