Escape Velocity

Using the GeoGebra app above, I intend to demonstrate the relationship between total energy, kinetic energy and gravitational potential energy in a rocket trying to escape a planet’s gravitational field.

By changing the total energy of the rocket, you will increase the initial kinetic energy, thus allowing it to fly further from the surface of the planet. The furthest point to which the rocket can fly can be observed by moving the slider for “distance”. You will notice that the furthest point is where kinetic energy would have depleted.

Gravitational potential energy of an object is taken as zero at an infinite distance away from the source of the gravitational field. This means gravitational potential energy anywhere else takes on a negative value of $\dfrac{-GMm}{r}$. Therefore, the total energy of the object may be negative, even after taking into account its positive kinetic energy as total energy = kinetic energy + gravitational potential energy.

The minimum total energy needed for the rocket to leave the planet’s gravitational field is zero, as that will mean that the minimum initial kinetic energy will be equal to the increase in gravitational potential energy needed, according to the equation $\Delta U = 0 – (-\dfrac{GMm}{R_P})$, where $R_P$ is the radius of the planet.

Since $\dfrac{1}{2}mv^2 = \dfrac{GMm}{R_P}$, escape velocity, $v = \sqrt{\dfrac{2GM}{R_P}}$.

Template for self-assessment questions

Here is a template that I might use to generate questions for students’ self-assessment in future. Based on a query that one of the participants in a GeoGebra online tutorial asked about generating random questions for simple multiplication for lower primary students.

The online tutorial was conducted by some teachers in the Singapore MOE GeoGebra community to share how GeoGebra could be used to create resources for home-based learning.

Angular velocity

This GeoGebra app shows how angular velocity ω is the rate of change of angular displacement (i.e. $\omega=\dfrac{\theta}{t}$) and is dependent on the speed and radius of the object in circular motion (i.e. $v=r\omega$).

Students can explore the relationships by doing the following:

Keeping r constant and varying ω.

Keeping ω constant and varying r.

Keeping v constant by varying r and ω.

Angular displacement

This GeoGebra app shows the relationship s = .

One activity I get students can do is to look at the value of θ when the arc length s is equal to the radius r. This would give the definition of the radian, which is the angle subtended at the centre of a circle by an arc equal in length to its radius.

Mathematics defines the constant π as the ratio of a circle’s circumference to its diameter. This can also be shown in the app, although you need to drag the moving point to a point just short of one full revolution.

Creating a simple interactive using GeoGebra

While preparing to share with some fellow teachers in Singapore about the use of GeoGebra in Physics, I came up with a set of simple instructions to create an interactive, while introducing tools such as sliders, checkboxes (along with boolean values) and input boxes. Download it here.

You should be able to follow the instructions in the pdf document above and make a simple interactive applet yourself too. The outcome of the interactive applet will be like this:

Using Loom and GeoGebra to explain a tutorial question

It’s Day 1 of the full home-based learning month in Singapore! As teachers all over Singapore scramble to understand the use of the myriad EdTech tools, I have finally come to settle on a few:

  1. Google Meet to do video conferencing
  2. Google Classroom for assignment that requires marking
  3. Student Learning Space for students’ self-directed learning, collaborative discussion and formative assessment.
  4. Loom for lecture recording
  5. GeoGebra for visualisation

The following is a video that was created using Loom to explain a question on why tension in a rope on which a weight is balanced increases when the rope straightens.