# Motion

#### Cisco Qid

##### Active member
Can the above explain an anomaly I've never understood?

A car drives at 100 mph into an immovable object. You can imagine the mess the car is in.

Another car drives at 50 mph into an immovable object. Obviously, the car is a mess - but nowhere near as bad a mess as the 100 mph car.

Two cars drive straight at each other at 50 mph. Relative to each other, they are travelling at 100 mph. When they collide, will the damage to the cards be approximately equivalent to the 100 mph car above, or the 50 mph car above?
The laws of conservation of momentum and energy state that they will stay the same before and after both collisions. Part of the momentum and energy is converted into heat which is in effect molecular velocity in the bodies involved. One of the details that you left out was the masses of both cars. For instance, if the mass of the car in the first incident is twice the mass of the other car then its momentum would be p1 = m1 x 100 = 2 x m2 x 100 = m2 x 200 and the momentum of the other car is p2 = m2 x 50 so that the momentum of the first car is four times greater than the second when hitting the wall and all that extra energy has to go somewhere both into heat and damage. In the second case with each other you have, p1 = m1 x 50 = 2 x m2 x 50 = m2 x 100 for the first car and p2 = -50 x m2 which is half the momentum of the first car but in the opposite direction. After the collision the net result is 50 x m2 in the direction of the bigger car minus any energy loss during the collision with the end result better determined by experimentation with possibly slightly differing results each time. But then again this is not a relative motion problem because if involves acceleration or rather deceleration.

#### LifeIn

##### Well-known member
Can the above explain an anomaly I've never understood?

A car drives at 100 mph into an immovable object. You can imagine the mess the car is in.

Another car drives at 50 mph into an immovable object. Obviously, the car is a mess - but nowhere near as bad a mess as the 100 mph car.

Two cars drive straight at each other at 50 mph. Relative to each other, they are travelling at 100 mph. When they collide, will the damage to the cards be approximately equivalent to the 100 mph car above, or the 50 mph car above?
The damage would be like the 50 mph case. The reason is that the 100 mph damage is only possible when hitting an immoveable object - the frame of reference being that immoveable object. But when we say the two cars have a relative speed of 100 mph, that is measuring each car with respect to the frame of reference of the other car. So let's pick one of the two cars as the frame of reference and look at the other car with respect to that frame of reference. The other car does indeed look like it is moving at 100 mph. But in this frame of reference, the first car no longer appears immoveable. In fact, immediate after contact, the other car (which was previously at rest in this frame of reference) is now moving backward at 50 mph. It is clearly moveable in this frame of reference. The only frame of reference in which the collision appears to be with an immoveable object is the frame of reference of the road upon which both cars are travelling at 50 mph. One of the lessons of relativity is that all measurements of speed must be with respect to a stated frame of reference. As long as we are careful to observe the frame of reference for each measurement, we will not fall into this type of paradox. (Even though this particular paradox does not involve Einsteinian relativity).

• Nouveau

#### Electric Skeptic

##### Well-known member
The damage would be like the 50 mph case. The reason is that the 100 mph damage is only possible when hitting an immoveable object - the frame of reference being that immoveable object. But when we say the two cars have a relative speed of 100 mph, that is measuring each car with respect to the frame of reference of the other car. So let's pick one of the two cars as the frame of reference and look at the other car with respect to that frame of reference. The other car does indeed look like it is moving at 100 mph. But in this frame of reference, the first car no longer appears immoveable. In fact, immediate after contact, the other car (which was previously at rest in this frame of reference) is now moving backward at 50 mph. It is clearly moveable in this frame of reference. The only frame of reference in which the collision appears to be with an immoveable object is the frame of reference of the road upon which both cars are travelling at 50 mph. One of the lessons of relativity is that all measurements of speed must be with respect to a stated frame of reference. As long as we are careful to observe the frame of reference for each measurement, we will not fall into this type of paradox. (Even though this particular paradox does not involve Einsteinian relativity).
Hm...your explanation is a little more understandable to me than Cisco Qid's (not that I am disparaging Cisco's - it was just a little over my head. Thanks to both of you for your answers - clearly both of you know vastly more physics than I). I think I might almost understand it.

#### Nouveau

##### Well-known member
Hm...your explanation is a little more understandable to me than Cisco Qid's (not that I am disparaging Cisco's - it was just a little over my head. Thanks to both of you for your answers - clearly both of you know vastly more physics than I). I think I might almost understand it.
Case A: Single car hits immovable wall at 50mph = 50mph worth of damage absorbed by one car.

Case B: Single car hits stationary car at 50 mph = 50mph worth of damage absorbed by two cars.

Case C: Single car hits stationary car at 100 mph = 100mph worth of damage absorbed by two cars.

Case D: Two cars hit each other head on at 50mph each (combined closing speed of 100mph) = 100mph worth of damage absorbed by two cars.

Case B will obviously involve less damage to the moving car than A. Case C should be the same damage per car as in A, with twice the momentum but also twice the material (two cars) absorbing the impact. Case D is exactly the same as C, only described from a different frame of reference, so should be the same damage per car as both C and A.

That was my initial reasoning, but this is obviously wrong. Hitting a wall at 50 is far more than twice as bad as hitting the same wall at 25. It's not momentum (m x v) that determines damage, but rather the kinetic energy (m x v^2). So double the speed is a lot more than just double the damage. It's exponential. So Case B is half the damage (per car) as A (same energy, but two cars to absorb it), while case C will be more damage per car than A (two cars absorbing the impact, but much more than twice the energy). And Case D remains the same as Case C.

So I think the answer to your question is neither. The damage per car in the head-on crash of Case D will be twice that of hitting an immovable wall at the same speed (of 50mph), yet still only half the damage of hitting an immovable wall at twice the speed (i.e. 100mph).

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#### inertia

##### Super Member
If motion is relative how can we know that we have accelerated rather than everything else slowing down?
Frictionless motion is particularly fun to watch while paying attention to the conservation of angular momentum. L = Iω

#### inertia

##### Super Member
Frictionless motion is particularly fun to watch while paying attention to the conservation of angular momentum. L = Iω

^This^ shows predicted motion with precision without the necessity of invoking special or general relativity.