Projectile Motion

The Trajectory

plotting the parabola

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

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 m/s t,y=10 m/s t−5 t2x = 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)(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)=(10,0)(x, y) = (\hl{10}, \hl{0})
The path curves: a parabolaThe ball sampled every half second, rising then falling along a curved path back to the ground.t=0t=1ball

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.

tvyy0 s10 m/s0 m1 s10 m/s5 m2 s10 m/s0 m\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}
The path curves: a parabolaThe ball sampled every half second, rising then falling along a curved path back to the ground.t=0t=1ball
mechanics With clean components and g = 10, the second-by-second positions are whole numbers that fall on a parabola.