Graphs of Vertical Projectile Motion

We can describe vertical projectile motion through a series of graphs: position, velocity and acceleration versus time graphs. This isn't new, in previous lessons, you learnt about the graphs for describing rectilinear motion with constant acceleration. Therefore, everything you learnt about the graphs describing rectilinear motion is applicable when describing vertical projectile motion.

Important:

Remember that: \[\vec{a}=\frac{\Delta \vec{v}}{\Delta t}\] \[\vec{v}=\frac{\Delta \vec{x}}{\Delta t}\]For the graphs of these quantities the slope and the area under a graph can tell us about changes in other quantities. As a reminder, here is a summary table, from Grade 10 rectilinear motion, about which information we can derive from the slope and the area of various graphs:

Graph	Slope	Area
Displacement vs time	\(\vec{v}\)	-
Velocity vs time	\(\vec{a}\)	\(\Delta \vec{x}\)
Acceleration vs time	-	\(\vec{v}\)

Stationary object
Uniform motion
Constant acceleration

The table above shows position vs time, velocity vs time and acceleration vs time graphs.

The graphs are the graphical representations of the equations of motion. This means that if you have the graph for one of position, velocity or acceleration you should be able to write down the corresponding equation and vice versa.

The type of graphs is related to the equation of motion: position is a parabola, velocity is a straight line and acceleration is a constant. The characteristic features then depend on the sign convention and the specific values of the problem.

To illustrate the variation, we consider three separate cases: (1) the object has an initial velocity in the upward direction (opposite direction to \(\vec{g}\)), (2) the object falls from rest (no initial velocity), and (3) the object has an initial velocity in the downward direction (same direction as \(\vec{g}\)).

Important: the horizontal axis in these graphs is time, not space.

Case 1: Initial velocity upwards

Consider an object undergoing the following motion:

at time \(t=\text{0}\text{ s}\) the object has an initial position, \(\vec{x}_i=0\), and an initial upward velocity of magnitude \(v_i\),
the object reaches a maximum height, \(h_m\), above the ground,
the object then falls down to a final position, \(\vec{x}_f\), with a final velocity \(v_f\) downward at time = \(t_f\).

The equation for position is given by: \(\vec{x} = \vec{x}_i + \vec{v}_{i}t+\frac{1}{2}\vec{g}{t}^{2}\). We have to choose a positive direction as before. We choose upwards as positive. The position versus time graph for the motion described is shown below:

Important:

It is important to notice that we are plotting position versus time and the initial position, \(\vec{x}_{i}\), is not necessarily zero. The position as a function of timewill be given by \(\vec{x}_f = \vec{x}_{i} + \vec{v}_{i}t+\frac{1}{2}\vec{g}{t}^{2}\).The initial position, \(\vec{x}_{i}\), will shift the graph of \(\vec{x}\)up or down by a constant value. Whether it is an upward or downward shift will depend on which directionwas chosen as the positive direction. The magnitude of the shift will depend on the choice of origin for thecoordinate system (frame of reference).

The second graph is a linear function showing the velocity as a function of time: \(\vec{v}_{f} = \vec{v}_{i}+\vec{g}t\). Note that we chose upwards as our positive direction therefore the acceleration due to gravity will be negative. The slope (i.e. the coefficient of \(t\)) of the velocity versus time graph is the acceleration and the slope is negative which is consistent with our expectations after choosing upwards as the positive direction.

The final plot is a graph of acceleration versus time, and is constant because the magnitude and direction of the acceleration due to gravity are constant.

Cases 2 and 3: Initial velocity zero or downwards

We start by choosing a direction as the positive direction. We want to compare the graphs with the first case, so we will choose the same direction (upwards) to be the positive direction.

The one thing that remains the same is the acceleration, \(\vec{g}\), due to gravity. Therefore:

If an object starts from rest (for example is dropped), then the initial velocity is zero. We know that there is constant acceleration and hence the graph of velocity versus time looks like this:

If the object starts with an initial velocity downwards the velocity graph looks like:

Important:

It is important to notice that the slope of all the velocity versus time graphs has the same magnitude. If the same direction is chosen as positive then the slope has the same magnitude AND the same sign (direction). The difference in these graphs is related to the initial velocity, \(\vec{v}_{i}\). The magnitude of the slope of the graph is the magnitude of the acceleration.

For the case where the object falls from rest, the equation for position is \(\vec{x} = \vec{x}_i +\underbrace{\vec{v}_{i}t}_{\text{0}}+\frac{1}{2}\vec{g}{t}^{2}\) and the graph of the position versus time looks like this:

For motion with an initial velocity in the negative direction, the graph of displacement versus time will be a narrower parabola as compared to the case where the object falls from rest. This is shown in this graph where the initial velocity is in the negative direction:

The primary difference in features between the cases we've looked at is that for Case 1, the position graph has a maximum positive displacement (it has a peak), while in Cases 2 and 3, where the initial velocity is negative, the displacement is always negative.

You may be wondering if the final position is always the same as the initial position, as we had for our Case 1 example. That is not necessarily true. If an object is thrown up from the edge of a cliff, the initial position of the object will be the height of the cliff but the final position will be at the base of the cliff. If we choose the point from which the object is thrown as the origin of our coordinate system then the graph will look like this:

If we choose to centre our coordinate system at the base of the cliff then the graph will look like this,because the initial displacement is the height of the cliff:

Graphs of Vertical Projectile Motion