The Trajectory - Mechanics Step by Step

Reading the across and up positions at each second and plotting the points traces the curved path of a thrown ball.

mechanics With clean components and g = 10, the second-by-second positions are whole numbers that fall on a parabola.

Example

Reading the across and up positions at each second and plotting the points traces the familiar curved path of a thrown ball. Exact arithmetic here means exact results for the stated model inputs; measured inputs still carry uncertainty and significant-figure limits.

highlighted = computed this step

Track both parts each second

We already have across position equal to 5 times time, and up position equal to 10 times time minus 5 times time squared. Now read both at each second.

x = 5\ \text{m}/\text{s}\,t, \quad y = 10\ \text{m}/\text{s}\,t - 5\,t^{2}

After one second

At one second the ball has moved across to 5 and risen to 5 metres. It is still climbing.

(x, y) = (5, 5)

After two seconds

At two seconds it has coasted across to 10 and fallen back to the ground at height 0. Plotting the points traces a parabola.

(x, y) = (\hl{10}, \hl{0})

The height values make the curve

The sideways part moves evenly, but the height values do not. That uneven vertical list is why the plotted path bends instead of forming a straight diagonal.

\begin{array}{c|c|c}t & v_y & y \\ \hline 0\ \text{s} & 10\ \text{m}/\text{s} & 0\ \text{m} \\ 1\ \text{s} & 10\ \text{m}/\text{s} & 5\ \text{m} \\ 2\ \text{s} & 10\ \text{m}/\text{s} & 0\ \text{m}\end{array}