As one of the first topics in A-level physics, kinematics introduces JC students to the variation of velocity and displacement with acceleration. Very often, they struggle with the graphical representations of the 3 variables.
This Geogebra app allows students to vary acceleration (keeping it to a linear function for simplicity) while observing changes to velocity and displacement. Students can also change the initial conditions of velocity and displacement.
The default setting shows an object being thrown upwards with downward gravitational acceleration of 10 m s-2.
The movement of the particle with time is shown on the left with a reference line showing the position on the displacement graph.
The following simulation allows users to observe the effect of air resistance on a parachutist before and after he opens his parachute. Try to open the parachute when the man first reaches terminal velocity and observe the changes in velocity.
This is a simulation done from scratch after attending a crash course on EJSS. It displays the velocity and acceleration vectors as an object is projected vertically upward with a uniform downward acceleration. The initial upward velocity can be altered using a slider. The graph of velocity is traced as the ball moves.
The following is a question (of a more challenging nature) posed to JC1 students when they are studying the topic of kinematics.
A gun is aimed in such a way that the initial direction of the velocity of its bullet lies along a straight line that points toward a coconut on a tree. When the gun is fired, a monkey in the tree drops the coconut simultaneously. Neglecting air resistance, will the bullet hit the coconut?
Two-Dimensional Kinematics: Gun and Coconut Problem
It is probably safe to say that if the bullet hits the coconut, the sum of the downward displacement of coconut $$s_{yc}$$ and the upward displacement of the bullet $$s_{yb}$$ must be equal to the initial vertical separation between them, i.e. $$s_{yc}+s_{yb}=H$$
This is what we need to prove.
Since $$s_{yc}=\frac{1}{2}gt^2$$
$$s_{yb}=u\text{sin}\theta{t}-\frac{1}{2}gt^2$$ and $$s_{xb}=u\text{cos}\theta t$$
At the same time, the relationship between $$H$$ and the horizontal displacement of the bullet $$s_{xb}$$ before it reaches the same horizontal position of the coconut is $$\text{tan}\theta=\frac{H}{s_{xb}}$$
I’ve used the open-source Tracker software, a video analysis and modeling tool built for use in Physics education, for both my IP3 and JC1 classes this year. Thanks to Mr Wee Loo Kang and his team for enthusiastically introducing this software to the physics teachers of Singapore.
For the IP3 cohort, students were tasked to analyse the movement of any sports-related projectile and to relate the variations in displacement, velocity and acceleration to one another in both dimensions. This was a direct transfer task for the topic of two-dimensional kinematics that they were taught in class. Attempts to explain these variations using the idea of forces were encouraged as well, even though that topic has not be formally introduced yet.
For my JC1 class, I explained the specific example of the bouncing ball using the software, which was useful to show the variation in vertical displacement, velocity and acceleration synchronously with the positions of the ball. I used the resources in the Singapore Tracker Digital Library, search for the following directories: 02_newtonianmechanics_2kinematics > trz > Balldropbounce4x.trk.
Use of Tracker to explain the bouncing ball graphs
It was easier for students to compare the three stages of the movement, namely Stage A: the way up, Stage B: the bounce (during which the ball is in contact with the ground) Stage C: and the way down.
A series of guiding questions such as the following will be useful:
Is there a difference between the vertical accelerations in stages A and C?
What do the gradients in the velocity time graph for stages A, B and C represent?
Identify the turning point of the ball in all 3 graphs. Notice that the acceleration remains at about 10 m s-2.
How would the graphs look like if the coordinate system / sign convention is changed such that the displacement is defined as zero at the floor and upward is taken to be positive? The effect of this change can be shown by dragging the axes on the video (the two perpendicular purple lines) to the bottom and rotating the horizontal axis 180o; by dragging the short purple line near the intersection.
Graphs with different coordinate system defined.
The following video (sorry, no audio) shows the steps to take to do all the above. Just pause it at any point and rewind if you didn’t catch what I did.
A video tutorial on the use of the Addestation datalogger with its motion sensor to measure the displacement of a bouncing ball and to observe the velocity and acceleration using its differentiation function.
