Getting a useful sketch out of a mechanics problem

Consider this:

A pirate drags a heavy treasure chest across a beautiful flat beach to take back to his ship. This savvy pirate happens to know that he is pulling with a force of 10N, and the chest has a mass of 20kg. He experiences a constant resistive force of 2N due to the friction of the sand. What is the acceleration of the treasure chest?

This is a stripped-back example of some real-life mechanics in action. It is pretty easy to visualise what is going on from the context, and we can build an appropriate model to do the calculation required.

Where to start

First things first, the acceleration of the treasure chest is our end goal. This is our object in motion that we care about. Let’s get a (basic) sketch together for this to validate that all the maths we throw at the paper is in the correct place. Start with a rectangle, our chest, on a flat surface, our beach.

We have a few numbers surrounding the chest: 10N force pulling it (courtesy of our pirate), and resistive force of 2N thanks to friction from the sand. Friction is going to work against the pirate pulling the chest (which we’ve explicitly been told is parallel to the beach itself).

Next we’re told the mass of the chest, 20kg. Therefore, the weight acting down by the chest is 20gN. This leads us to the resultant force of the chest, R, which will just be equal to the weight since the chest is moving neither up or down.

Great, now the chest will be accelerating, 𝒂𝑚𝑠⁻² in the direction of the greatest horizontal force acting on it, which is coming from the direction the pirate is pulling.

And our illustration is complete! Everything is in front of us apart from a few mechanics equations that you shall know like the back of your hand by the time the exam comes around.

Feel ready to give the question a shot? Check out the solution below once you’ve had a go!

Inspiration for this pirates / maths crossover: Source

Resolve horizontally to find the net horizontal force

Now recall Newton’s second law of motion: the force needed to accelerate a particle is equal to the product of the mass of the particle and the acceleration produced. This is more simply put as

Cool, our acceleration in question is in the same direction as the force we are moving by, R, so that’s a good mental check to have!

Rearrange to find 𝒂,

We can put this back into the context of our problem and say that the treasure chest is accelerating at 0.4𝑚𝑠⁻² across the beach

The big picture

Being able to make this simple sketch opens up a lot of doors later in the mechanics papers as you are able to reassure yourself that you are using the maths correctly. It’s all good news that you know the equations to solve the problem, but if you are unsure about which directions your forces are working without a clear sketch then you can open yourself up to some missed marks.

Where to next?

Fancy checking out some more contextual mechanics problems like this? We’ve got you covered over at AITutor, including worked solutions, for both AS and A2!

Here’s a tasty A2 moments question and sketch to get you intrigued:

A caveman walking up a hill drops a 7kg watermelon. The watermelon rolls down the slope reaching a speed of 15.36 𝑚𝑠⁻¹ after travelling for 7s. If the hill makes an angle of 24° to the horizontal, what is the coefficient of friction of the watermelon?

See if you can recreate the sketch below!