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I am currently trying to model a function for the 2d projectile motion of a football from the angle of elevation and the initial speed, however I am a bit confused on what the graph of projectile motion represents exactly.

Almost every textbook I have investigated gives the graph either without labels or with contradictory labels and it is a bit confusing, for example

Graph

The label for the Y axis on this graph seems to be position, but it is not stated whether it is the horizontal position or the vertical position.

Furthermore in many examples the range of the horizontal displacement is shown on the same graph further confusing things

Graph 2

This seems to show both the vertical and horizontal components of displacement, however wouldnt there need to be an additional axis for time in this case?

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The vertical axis for a projectile motion graph is generally always vertical displacement/height/vertical position, etc.

Both a lower axis in time and a lower axis in horizontal displacement are valid choices. You can use either one.

The lower axis in time represents the rise and fall of the body in the vertical axis of displacement, making no statement about its horizontal motion. It is described completely by the one-dimensional SUVAT equation $s_y(t) = u_y t + \frac{1}{2}gt^2$, where $s_y$ is the vertical displacement (I have chosen up to be the positive direction), $u_y$ is the initial vertical velocity, and $g = -9.81\mathrm{\,m\,s^{-2}}$.

The lower axis in horizontal displacement means that your 2-D graph represents a slice of the real world - the x-y plane in which the ball is moving - and shows the path through which the ball would move. If you were able to print out that graph at real size, with one metre on the axes being one metre in real life, and hang it to your side such that it appeared behind the projectile as you threw it, the projectile would fly through the air following the plotted parabola.

You would not be able to do that with a vertical displacement-time graph, unless your time scale was converted to metres by multiplying by the horizontal velocity, in which case you effectively have a trajectory graph again.

Note that in projectile motion, the horizontal velocity is modelled as a constant - which means that the horizontal displacement is directly proportional to the time in flight, and vice versa. As such, both graphs will always be parabolas, with one being a horizontal stretch transformation of the other.

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The label for the Y axis on this graph seems to be position, but it is not stated whether it is the horizontal position or the vertical position.

$h_m$ is mentioned near y-axis that simply implies it is the height in meters as a function of time from horizontal displacement $x_i$ to $x_f$(As mentioned on the x axis).

There is no specific "Graph of Projectile Motion". Either it is trajectory(the 2nd graph) or height-time graph(the 1st graph). The graph chosen in purely based on the information the author wants to convey or want to talk about.

For sure you can represent position as a function of time by adding time axis to 2nd graph but practically it only complicates things which is neither benefitial for understanding the motion nor for problem solving.

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For 1D projectile problems, what's usually represented is the $y$ position or displacement vs. time. The graph will be a parabola.

The equation plotted is $$y-y_0=-\dfrac12gt^2+v_0t$$

For 2D projectile motion, there are usually two common scenarios.

Trajectory plot. This is the $y$ position vs. $x$ position. Time is not on this plot (though it can be extracted from the plot with some algebra). This plot will be parabolic (assuming 2D motion). If the projectile is launched from coordinate $(0,h)$, the equation of the $y$ vs. $x$ position is:

$$y = -\dfrac{g}{2v_0^2 \cos^2\theta} x^2 + x\tan\theta + h$$

The above equation gives you the path (i.e. trajectory) that the projectile will take.

Position vs time plots. These are plots representing the position as a function of time. Since you have both $x$ and $y$ positions, you can make two graphs:

  1. Graph for $x$ position vs. time $t$. Namely, $x-x_0=v_{0x}t$
  2. Graph for $y$ position vs. time $t$. Namely, $y-y_0=-\dfrac12gt^2+v_{0y}t$

Notice how the 1D $y$ vs. $t$ plot is (almost) identical to case 2.

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