Safety in Vehicles

The safety of the traveller and staff is of paramount importance to every responsible operator.  Everything possible is done to prevent vehicles coming into contact with other vehicles.  However sometimes the worst does happen and vehicles are involved in some kind of impact.

In the old days it was thought that both cars and railway carriages should be made as strong as possible to withstand impacts.  However experience showed that people suffered even more serious injuries, despite the fact that the vehicles suffered little damage.  One of the most serious aspects of collisions in railway vehicles was the effect of telescoping, where the solid and very strong under-frames of the coaches would ride over each other, crushing the bodywork (and anyone in the way).

In the present day both cars and railway carriages have crumple zones to absorb much of the impact.  In the event of an accident they are more seriously damaged, but the people inside can get up and walk away.  A bent railway carriage can be repaired or replaced; a person cannot.

We will look at collisions that occur between trains that are on the same track.  They shouldn't be, but they are.

Momentum in Collisions

In any collision, the impact forces:

• are equal and opposite (Newton III - forces act in pairs);

• last for equal times.

This may or may not result in equal damage; that's immaterial.

Also in any collision the momentum is conserved.  Momentum is the product of mass and velocity:

momentum (kg m s^-1) = mass (kg) × velocity (m s^-1)

In Physics code:

p = mv

Momentum is a vector quantity.  Its correct use is essential in understanding collisions.  In every collision momentum is conserved:

momentum before = momentum after

The impulse (change in momentum) in a collisions is given by:

Dp = FDt

We will consider collisions in terms of railway wagons being shunted about sidings.  There is an operation called fly-shunting in which wagons are pushed by an uncoupled shunting locomotive.  The locomotive is stopped and the wagon is allowed to roll freely and runs into the other wagons on the siding.  Too slow and the wagon comes to rest short of the other wagons.  Too fast and the wagon will impact violently into the other wagons and may derail or be damaged.  It is less common practice nowadays, and is forbidden on many railways.

Consider two wagons, wagon 1 and wagon 2 of mass m1 and m2 respectively.   Before the collision they have velocities u1 and u2.  After the collision they have velocities v1 and v2.  We will consider the velocities being positive if they are from left to right.  This is summed up in the table, plus some numbers, which we will use in worked examples.

Here is the first situation.  Wagon 1 is travelling at +u1 m/s, while wagon 2 is stationary.  The two wagons stick together.

(Rail buffs among you will recognise that these wagons are rather dated (Who said Thomas the Tank Engine?), but the physics is just as valid!)

Momentum before = momentum after

m1u1 + 0 = (m1 + m2)v

Now let us consider that both wagons are moving in the same direction:

For there to be a collision, wagon 1 must be travelling faster than wagon 2.  So the momentum can be described as:

m1u1 + m2u2 = m1v1 + m2v2

Now let's look at an example where the signs are important.  Remember that from left to right is positive and right to left is negative:

Momentum in Explosions

An explosion not just where something goes BANG.  In physics it is any situation where there is zero momentum at the start.  Since linear momentum is conserved, it means that the total momentum at the end must be zero.  Consider our two wagons that this time are buffered up very tightly and the springs in the buffers are ready to push them apart.

When the wagons are released, they fly apart in opposite directions as in the picture below:

Since momentum is conserved and the momentum at the start was zero, we can write:

0 = m1v1 + m2v2

which we can rearrange to:

m1v1 = -m2v2

Impulse

We are familiar with the relationship that sums up Newton's Second Law:

F = ma

If we consider the make up of this statement we can write:

Force = mass × acceleration

Force = mass × change in velocity

                    time interval

It is very reasonable to write that the product of mass × change in velocity is the change in momentum.  We can therefore write:

Dp = mDv

So we can rewrite Newton II as:

F = Dp

     Dt

We can rearrange the equation to give:

Dp = FDt

The term Dp is the change in momentum or impulse.  We can also consider the impulse to the product between force and time interval.  Therefore impulse can have the units newton seconds (N s) which is the same as kilogram metres per second (kg m s^-1).

A big force applied for a short time interval will have the same impulse as a small force applied for a long time interval.

The area under this graph is the impulse.  You can see that the area of both rectangles is the same.  The significance for vehicles is that we want to increase the time, so that the force involved in a given impulse is made less.  Crumple zones in railway carriages and cars achieve this.

Energy in Collisions

Most collisions involve a loss of kinetic energy.  We can measure the kinetic energy before and after the collision, and we mostly find that the total kinetic energy is reduced.  The lost energy is turned into noise, heat, and work done to bend the structure.

Note that in this example we have ignored the signs.  This is because the negative velocities are squared, so they become positive.  And this is why energy is a scalar.

The loss of energy in this case is quite considerable.  At the speeds quoted, no doubt the wagons would be seriously damaged.  We can say that this is not a perfectly elastic collision.  It is a plastic collision.  In some books you will read about inelastic collisions.  A perfectly inelastic collision is where the objects stick to each other.

An elastic collision occurs when there is zero loss of kinetic energy.  Colliding objects store the energy as elastic energy, transferring back to kinetic energy.  We can write this as:

Ek1 (before) + Ek2 (before) = Ek1 (after) + Ek2 (after)

Perfectly elastic collisions occur when atoms bounce off each other.  However in the context of collisions of vehicles, they are decidedly undesirable.  However in sports activities, we want the collisions between a racquet and a ball to be nearly elastic, so that as little kinetic energy is lost as possible.