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February 11

Collision analysis

A bus of mass 25 tons starts to travel along a straight road from A to B at 50 mph. At the same time a bee of mass 2.5 grams starts to fly from B to A along the same road at 5 mph. Eventually the bee and the bus collide.

Assume the bee impacts a surface of the bus which is flat and at right angles to its direction of travel, such as the windscreen, and the bee sticks to this surface.

Then the bus will be slowed by a tiny amount reflecting the relative masses / momenta of the bee and bus. In addition, relative to A, the velocity of the bee changes from -5 mph to +50 (approximately) mph during the collision. So at some stage during the collision the bee must be stationary. This can only be when the bee is in contact with the bus.

The question is : Why is the bus not stationary at the same time?

Does it make any difference to the situation if the bee is replaced by either a perfectly elastic object, or a perfectly rigid object, of the same mass?

And does it make any difference if the collision is considered at the level of the individual atoms involved? Ionlywanttoknow (talk) 18:40, 11 February 2024 (UTC)[reply]

If you define the direction from A to B as positive, the bee is changing from a speed of -5 mph to +50 mph. Regardless of the exact nature of that change, in order to get from -5 to +50, at some point the speed must pass 0.

The bus's speed is changing from +50 mph to +49.999999... mph. That change does not pass through zero, so the bus is never stationary.

You could get into more details of the interaction, but that's the basic point. PianoDan (talk) 20:11, 11 February 2024 (UTC)[reply]

There's two explanations. If you're thinking of absolutely rigid bodies then the bee is never stationary, it immediately switches between going in one direction and the other. In the physical world there is always an bit of elasticity so the bus atoms can keep going forwards whilst the bees ones slow down and reverse as the bee is squashed against the windscreen. NadVolum (talk) 20:30, 11 February 2024 (UTC)[reply]

This follows from the Intermediate value theorem. Ruslik_Zero 20:35, 11 February 2024 (UTC)[reply]

There is no intermediate value of speed in the case of rigid bodies. NadVolum (talk) 21:06, 11 February 2024 (UTC)[reply]

Here is the argument: The intermediate value theorem tells you that the speed of the bee is zero at some points in time. By assumption, the bee and the bus move at the same speed when they are in contact. So the bus must move at speed zero at that point in time. So far, so good. But we are talking about points in time. Speed, being distance per time, is not well-defined for an individual point. You need an "expanse of time" to have speed. --Stephan Schulz (talk) 10:20, 12 February 2024 (UTC)[reply]

The intermediate value theorem is valid for continuous functions. For rigid bodies, the speed is discontinuous (acceleration infinite), so the IVT does not apply. As was said above, real bodies are always elastic to some point, which causes gradual changes of the speed and keeps the accelerations finite. --Wrongfilter (talk) 12:06, 12 February 2024 (UTC)[reply]

Why is the bus not stationary? Because it is moving at 50 mph! For the bus to decelerate from 50 mph to stationary in the brief duration of this collision would take a very, very large force. So large it could not possibly occur in a collision with a small object. Dolphin (t) 06:19, 12 February 2024 (UTC)[reply]

The meaning of "why" in the question is unclear. However, one of many ways to see that the bee+bus system cannot be stationary is that its kinetic energy would be zero, violating the law of conservation of energy. In the idealized analysis, the bee is treated as if it is a point particle. In a more refined analysis, the collision takes some time; the drama unfolds in about half a millisecond. Halfway through the process, the front part of the insect has already been squashed and splotched across the bus, co-moving with it at almost 50 mph. Shockwaves through its body may have caused parts of its now liquefied internals to erupt through its rear, moving even faster than the bus. Other parts of the bee are still moving in the original direction, towards the bus. If we zoom in to the elementary particle level, we run into the fundamental limit of quantum uncertainty: we cannot assert meaningfully that any part of the bee is stationary at any time. --Lambiam 13:30, 12 February 2024 (UTC)[reply]

By the way you might like to read Windshield phenomenon about why this hasn't been happening much recently. NadVolum (talk) 17:01, 12 February 2024 (UTC)[reply]

According to the reincarnated bee (murmuring something about local inertial frames not accelerating) it was the bus that stopped and the road reversed and took the bus+bee to the land of honey. Modocc (talk) 14:18, 13 February 2024 (UTC) [reply]