successive-collisions-in-1d

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 <img alt=”2 over 5 cross times 5 over 17 equals 2 over 17″ data-mathml='<math ><semantics><mrow><mfrac><mn>2</mn><mn>5</mn></mfrac><mo>×</mo><mfrac><mn>5</mn><mn>17</mn></mfrac><mo>=</mo><mfrac><mn>2</mn><mn>17</mn></mfrac></mrow><annotation encoding=”application/vnd.wiris.mtweb-params+json”>{“language”:”en”,”fontFamily”:”Times New Roman”,”fontSize”:”18″,”autoformat”:true}</annotation></semantics></math>’ height=”47″ role=”math” src=”data:image/svg+xml;charset=utf8,%3Csvg%20xmlns%3D%22http%3A%2F%2Fwww.w3.org%2F2000%2Fsvg%22%20xmlns%3Awrs%3D%22http%3A%2F%2Fwww.wiris.com%2Fxml%2Fmathml-extension%22%20height%3D%2247%22%20width%3D%22104%22%20wrs%3Abaseline%3D%2230%22%3E%3C!–MathML%3A%20%3Cmath%20xmlns%3D%22http%3A%2F%2Fwww.w3.org%2F1998%2FMath%2FMathML%22%3E%3Cmfrac%3E%3Cmn%3E2%3C%2Fmn%3E%3Cmn%3E5%3C%2Fmn%3E%3C%2Fmfrac%3E%3Cmo%3E%26%23xD7%3B%3C%2Fmo%3E%3Cmfrac%3E%3Cmn%3E5%3C%2Fmn%3E%3Cmn%3E17%3C%2Fmn%3E%3C%2Fmfrac%3E%3Cmo%3E%3D%3C%2Fmo%3E%3Cmfrac%3E%3Cmn%3E2%3C%2Fmn%3E%3Cmn%3E17%3C%2Fmn%3E%3C%2Fmfrac%3E%3C%2Fmath%3E–%3E%3Cdefs%3E%3Cstyle%20type%3D%22text%2Fcss%22%3E%40font-face%7Bfont-family%3A’math13d2dc549f508103e95d72be633’%3Bsrc%3Aurl(data%3Afont%2Ftruetype%3Bcharset%3Dutf-8%3Bbase64%2CAAEAAAAMAIAAAwBAT1MvMi7iBBMAAADMAAAATmNtYXDEvmKUAAABHAAAADxjdnQgDVUNBwAAAVgAAAA6Z2x5ZoPi2VsAAAGUAAABIWhlYWQQC2qxAAACuAAAADZoaGVhCGsXSAAAAvAAAAAkaG10eE2rRkcAAAMUAAAADGxvY2EAHTwYAAADIAAAABBtYXhwBT0FPgAAAzAAAAAgbmFtZaBxlY4AAANQAAABn3Bvc3QB9wD6AAAE8AAAACBwcmVwa1uragAABRAAAAAUAAADSwGQAAUAAAQABAAAAAAABAAEAAAAAAAAAQEAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAACAgICAAAAAg1UADev96AAAD6ACWAAAAAAACAAEAAQAAABQAAwABAAAAFAAEACgAAAAGAAQAAQACAD0A1%2F%2F%2FAAAAPQDX%2F%2F%2F%2FxP8rAAEAAAAAAAAAAAFUAywAgAEAAFYAKgJYAh4BDgEsAiwAWgGAAoAAoADUAIAAAAAAAAAAKwBVAIAAqwDVAQABKwAHAAAAAgBVAAADAAOrAAMABwAAMxEhESUhESFVAqv9qwIA%2FgADq%2FxVVQMAAAIAgADrAtUCFQADAAcAZRgBsAgQsAbUsAYQsAXUsAgQsAHUsAEQsADUsAYQsAc8sAUQsAQ8sAEQsAI8sAAQsAM8ALAIELAG1LAGELAH1LAHELAB1LABELAC1LAGELAFPLAHELAEPLABELAAPLACELADPDEwEyE1IR0BITWAAlX9qwJVAcBV1VVVAAIAgABVAtUCgAADAAcARhiwARQAsQAAExCxAAnksQABExCwBDyxBgj0sAI8MAGxCAETELEAA%2FawBzyxAQX1sAY8sgUHABD0sAI8sQkD5rEEBfWwAzwTMwEjETMBI4BVAgBVVf4AVQKA%2FdUCK%2F3VAAAAAAEAAAABAADVeM5BXw889QADBAD