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. $v$ is usually the speed after the first collision, so $w$ 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 $Speed = \frac{Distance}{Time}$
- or its rearrangements of $Speed \times Time = Distance$ and $Time = \frac{Distance}{Speed}$
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. $x$ can be found for $\frac{2}{5} = \frac{x}{\frac{5}{17} d}$ by finding $\frac{2}{5} \times \frac{5}{17} = \frac{2}{17}$ on your calculator, and then the answer would be $x = \frac{2}{17} d$
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 $v_{A}$ and $v_{B}$ are approaching each other, a distance $x$ apart
- They will cover distance $x$ , moving closer to each other at a rate of $(v_{A} + v_{B}) {ms}^{- 1}$ , 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 $v_{A} = 3$ and $v_{B} = 4$ then the ratio of distance covered will be $3 : 4$
  - So they will meet at a point $\frac{3}{7} x$ from A's starting point

Exam Tip

These questions can be difficult to visualise in your head so draw simple diagrams to show each collision.
Use common sense, and think how many possible (or impossible) ways there are for objects to move after the first collision. You will often have to consider the speed of one or more objects to decide if a second or third collision is possible.
These questions can involve lots of algebra, negatives and inequalities so do not rush them as you might make a silly mistake which can affect subsequent parts.

Worked example

Three uniform balls $A, B$ and $C$ of equal radius, and mass 0.1 kg, 0.2 kg and 0.3 kg respectively, can move along the same straight line on a smooth horizontal table with $B$ in the middle of $A$ and $C$ . $A$ and $B$ are projected towards each other in opposite directions with speed $10 m s^{- 1}$ and $2 m s^{- 1}$ respectively while $C$ is at rest. $A$ and $B$ collide directly which does not change the direction of motion of $A$ and subsequently $A$ moves with speed $1 m s^{- 1}$ .

Show that the speed of

B

immediately after it collides with

A

2.5 m s^{- 1}

In the subsequent motion,

B

collides directly with

C

. Immediately after this collision,

C

moves with speed

1.5 m s^{- 1}

. Determine if there will be a second collision between

A

and

B

4-1-2-collisions---multiple-example-solution

Worked example

Particles P and Q have masses 2kg and 3kg respectively. Initially they are moving in the same direction along the same straight line, such that the speed of particle P is $4 {ms}^{- 1}$ and the speed of particle Q is $1 {ms}^{- 1}$ . The two particles collide when they are at a distance d from a smooth fixed vertical wall, which is perpendicular to their direction of motion. After the collision with P, particle Q collides directly with the wall and rebounds so that there is a second collision between P and Q.
This second collision takes place at a distance x from the wall.

Given that the coefficient of restitution between P and Q is 1, and the coefficient of restitution between Q and the wall is 0.5, find x in terms of d.

worked example finding the distance of second collision from a wall

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

Revision Note