If you are a teacher then please also visit my new site: intermathematics.com for over 2000+ pdf pages of resources for teaching IB maths!

**Bullet Projectile Motion Experiment**

This is a classic physics experiment which counter to our intuition. We have a situation where 1 ball is dropped from a point, and another ball is thrown horizontally from that same point. The question is which ball will hit the ground first?

*(diagram from School for Champions site)*

Looking at the diagram above you might argue that the ball that is dropped falls to the floor quicker as it has a shorter path. Or, you might think that the ball thrown sideways would travel faster to the ground because of its initial horizontal velocity. Both of these views are wrong however – as both balls will land at exactly the same time. To understand why, let’s look at the 2 situations in turn.

**The ball launched sideways**

To show that both balls would hit the ground at the same time we need to split the motion into its x and y components. We have

Where the angle theta is the angle of launch, v is the initial velocity, g is the gravitational constant 9.8 m/s. If we have a launch from the horizontal direction, then this angle is 0, which gives the simplified equations:

x = vt

y = 0.5gt^{2}

if we relabel y as the vertical distance (d), then we have:

which is the time taken (ignoring air resistance etc) for an object launched horizontally to fall a distance d, where g is the gravitational constant 9.8 m/s.

So if we have a ball launched at a speed of 1 m/s from a height of 1m, it would hit the ground when:

t = (2/9.8)^{0.5} = 0.45 seconds

So we can use this value of t to see how far in the x direction it has travelled:

x = vt

x = 1(0.45)

x = 0.45m.

**The ball dropped vertically**

We still start with:

But this time we have no initial velocity as so we simply get:

x = 0

y = 0.5gt^{2}

or as before, if we relabel y as the vertical distance (d), then we have:

So with a ball dropped from a height of 1m, it would also hit the ground when:

t = (2/9.8)^{0.5} = 0.45 seconds

But this time the distance in the x direction will of course be 0.

**Showing this graphically **

We can also show this graphically using the tracker software. This allows you to track the motion of objects in videos. So using the video above we can set the axis, and the height of the table

and then the motion capture software actually plots the parabola of the ball’s motion.

This first graph shows the change in the y direction with respect to time for the ball launched horizontally. We have large steps because the video was in super slow motion, so there were frames of very little movement. Nevertheless we can clearly see the general parabola, with equation:

y = -0.43x^{2} -1.2x + 107

The second graph shows the change in y direction with respect to time for the ball dropped vertically down. As before we have a clear parabola, with equation:

y = -0.43x^{2} -1.2x + 106

Which is a remarkably close fit. So, there we go, we have shown that the vertical motion of our 2 objects are independent of their horizontal motion.

**Essential Resources for IB Teachers**

If you are a **teacher** then please also visit my new site. This has been designed specifically for teachers of mathematics at international schools. The content now includes over **2000 pages of pdf content** for the entire SL and HL Analysis syllabus and also the SL Applications syllabus. Some of the content includes:

**Original pdf worksheets**(with full worked solutions) designed to cover all the syllabus topics. These make great homework sheets or in class worksheets – and are each designed to last between 40 minutes and 1 hour.**Original Paper 3 investigations**(with full worked solutions) to develop investigative techniques and support both the exploration and the Paper 3 examination.- Over 150 pages of
**Coursework Guides**to introduce students to the essentials behind getting an excellent mark on their exploration coursework. - A large number of
**enrichment activities**such as treasure hunts, quizzes, investigations, Desmos explorations, Python coding and more – to engage IB learners in the course.

There is also a lot more. I think this could save teachers 200+ hours of preparation time in delivering an IB maths course – so it should be well worth exploring!

**Essential Resources for both IB teachers and IB students**

1) Exploration Guides and Paper 3 Resources

I’ve put together a **168 page** Super Exploration Guide to talk students and teachers through all aspects of producing an excellent coursework submission. Students always make the same mistakes when doing their coursework – get the inside track from an IB moderator! I have also made **Paper 3 packs** for HL Analysis and also Applications students to help prepare for their Paper 3 exams. The Exploration Guides can be downloaded here and the Paper 3 Questions can be downloaded here.

## 2 comments

Comments feed for this article

June 13, 2016 at 3:16 pm

BobIs it “velocity in the y-direction” or “y-component of the velocity”? Just semantics, and I know we all say both, but is the first really valid? The velocity is a single vector: There are not two velocity vectors, each pointing in orthogonal directions.

June 29, 2016 at 11:49 pm

CaliHi I would really appreciate it if you could help me with getting access to the motion capture software that you are using in this ME. Thanks a lot!