Optional Practical: suvat (GeoGebra)

Introduction

In this exercise the mathematical modelling programme GeoGebra will be used to show how the motion of a particle is related to the equations of motion and vice versa. First you will need to download and install GeoGebra, you can do that here.

The suvat equations are a set of equations that model the motion of a particle with constant acceleration. If a body starts with displacement so and velocity u then if it accelerates with acceleration a for a time t as represented in the diagram below. The displacement and velocity are defined by the following equations. These equations represent the motion but we will use them to create the motion in GeoGebra. Open GeoGebra and familairise yourself with the layout. Setting up the sliders

Before you can simulate the movement of a particle you must define the variables with sliders, first we shall define a slider for the initial displacement so .

• Select "slider from the tool bar and then click somewhere at the top of the graphics view. A box will open.
• Change the name of the slider from a to s_o (this will be so in the algebra view).
• Set the minimum to -10 and the max to 10 with increment 0.1 then "apply". Try sliding the slider, see how the value of so changes as you do so.

• Add sliders for initial velocity (u), acceleration (a) from -5 to 5 with increment 0.1 and time (t) from 0 to 5 increment 0.1.

The next stage is to use the equation s = so + ut + 1/2at2 to calculate the position at some time t.

• In the input bar type the equation s = s_o + u*t + 0.5*a*t^2 (note that you can use space instead of * for multiply)

You should see an equation the number s appear in the algebra pane. Vary a, so, t and u to see how s changes. You could make sure everything has been done correctly by solving this problem with your calculator and checking with GeoGebra.

A body starting with velocity 2 ms-1 travels with an acceleration of 4 ms-2. Calculate its displacement after 3 s

Now you are going to add a point that will move according to this equation.

• Add a point to the graphics view by clicking the point tool then click somewhere on the graphics view. You will see a point A on the graph and the coordinates of A listed in the algebra pane.
• Change the coordinates of the point (double click the coordinates) to (-5,s).
• Vary so, u, a and t to see how the point moves. Animating the point

Varying t at a constant rate will cause the point to move according to the equation for s. You can do this by sliding the slider or setting the properties of the slider to make it increase automatically.

• Right click the slider for t.
• "Tick" animation on.

This will make t increase and decrease. You don't want time to go backwards so adjust the animation settings.

• Right click the slider for t and select "object properties".
• Under animation select "increasing". • You can pause the animation with the pause button at the bottom left of the graphics view.

You can add an image to the point as shown below. The problem with using a rocket is that it always accelerates in the direction its travelling so to represent a negative acceleration it should be pointing downwards. This can be solved by using two images one with the rocket travelling up and the other down. To hide the unwanted image right click it and select "object properties" then under the "advanced" tab type a>0 in the "condition to show object" box. This image will now only show if a is positive. Drawing graphs

Displacement time graphs are often used to represent the motion of a body, you can make a point draw the graph by giving it coordinates (t,s).

• Add a new point to the graphics view.
• Change the coordinates of the point to (t,s).
• Animate t and observe the path of the point.
• Right click the point (might be best to pause he animation first) and select "trace on". Play the animation and observe the curve (poetic).

Instead of plotting the graph using a series of points it is also possible to draw the line representing the function. We want to plot s against t, with s on the y axis and t on the x axis. The equation relating s and t is s = so + ut + 1/2at2 to plot this we need to change s to y and t to x, this gives y = so + ux + 1/2 ax2

• Enter this equation in the input bar. You will see the line appear on the graphics view and the equation for the line in the algebra pane.
• Observe how the point follows the curve.
• You can move the axis around using the "move graphics view" tool or zoom in and out with your mouse wheel.

Note that the curve includes -ve values of time, this is before the clock was started.

• Try sketching the s-t curve for the following situations then use your simulation to see if you are right.

1.
Initial displacement = 5 m
Initial velocity = 2 ms-1
acceleration = 2 ms-2

2.
Initial displacement = -5 m
Initial velocity = -4 ms-1
acceleration = 2 ms-2

3.
Initial displacement = 0 m
Initial velocity = 2 ms-1
acceleration = -2 ms-2

Velocity time graph

You can't plot velocity vs time on the same axis as displacement time so you must first open a new window.

• Open a second graphics view by choosing graphics 2 from the view menu.
• We know that v = u + at so write an equation to calculate velocity.
• Add a point on graphics view 2 and define its coordinates so that it will plot the velocity vs time.
• Add an equation for the line representing velocity vs time.
• Try sketching velocity - time graphs for the situations above then use you simulation to see if you are right.

Vectors

Another feature of GeoGebra is the possibility to work with vectors, here we will simply use them to display the direction and magnitude of the different quantities. To do this a vector will be drawn joining two points that move with the moving point (or rocket if you bothered with that bit).

• Add a point just above the original point A in graphic view 1.
• define the coordinates of this point as (-5, s+a).
• Add a vector from A to the new point. The length of this vector will now be proportional and in the same direction as the acceleration.

• Make a second vector for velocity.
• To label the vectors a and v > right click > choose object properties > enter the correct letter as a caption > tick show label > choose caption.

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