Successive Collisions in 1D (Edexcel A Level Further Maths): Revision Note

Successive Collisions in 1D

Can there be successive/multiple collisions?

• After two objects collide it is possible that one (or both) of them collides with something else such as

  • A third object

  • A wall (perpendicular to the motion)

• Deal with each collision separately and use the steps for direct collisions

  • Drawing a separate diagram for each collision can help

  • Be clear and unambiguous with labelling and variables

    • e.g.  is usually the speed after the first collision, so  may be used for the speed after the second collision

Can there be a second collision between the original two objects?

• Let A, B and C be three objects travelling in the same straight line and suppose A and B collide directly and subsequently B and C collide directly

• After the collisions between A and B and B and C there will be a second collision between A and B if:

  • One is stationary and the other is travelling towards it

  • Both are travelling in opposite directions towards each other

  • Both are travelling in the same direction and the one in front is slower than the one behind

• The process is similar if object C is a wall

  • After B collides with the wall its direction will be reversed so it will be travelling towards A

  • B will collide with A again if its velocity in that direction is greater than the velocity of A in that direction (or if A was also travelling towards the wall after the first collision!)

  • To help you work out the speed of B after hitting the wall you will be given extra information such as the impulse exerted by the wall or the coefficient of restitution

How do I solve collisions questions involving distance and time?

• As momentum problems always deal with velocities, there may also be questions that involve distances, and times

• In between the collisions in collisions questions we are dealing with constant speed, so there is no need to use the suvat equations

  • We can instead make use of 

  • or its rearrangements of  and 

• In these types of questions, the distances and/or times will usually be algebraic

  • e.g. "The collision between A and B takes place a distance d from the wall"

  • A calculator can still be used to help with any complicated fractions, just remember to account for the algebraic term afterwards

  • e.g.  can be found for by finding  on your calculator, and then the answer would be 

• Keeping track of where the objects are, and when, in these problems is key

  • Draw a diagram for each stage

  • Use different letters for the final velocities of each stage e.g. v, and then w

  • Label distances on the diagrams

• A common mistake after two objects collide, is forgetting that they are both moving

  • e.g. If A and B collide, and then B goes on to collide with a wall, and rebounds to strike A again

  • A will no longer be in the location where A and B collided; it will have moved whilst B was travelling to the wall

    • To find how far A has moved, the time for B to reach the wall would need to be found first

    • This could then be used in conjunction with A's speed, to find how far it has moved in this time

• A common scenario in a part of these problems is when two objects at speeds  and  are approaching each other, a distance apart

  • They will cover distance , moving closer to each other at a rate of , this can be used to find a time for when they collide

  • The distance between the objects will reduce in proportion to their speeds

    • e.g. If  and  then the ratio of distance covered will be 

    • So they will meet at a point  from A's starting point