Estimate the initial velocity of the cars after the collision

[peace]

Homework Statement

In the event of a collision between two cars, pieces attached to them may be thrown into adjacent areas. How can the initial velocity of two cars be Estimated from such an observation? (Try a numerical example.)

Relevant Equations

I'm not sure about that.

What came to my mind for this question is:
Consider one of the cars. The velocity and mass of this car are V and M respectively.
And the velocity and mass of the piece attached to the car are m, v respectively.
Before the collision, the velocity of this piece relative to this car is zero. So its momentum is zero.
Before the collision, but the car has a non-zero velocity. So it has a non-zero momentum.
The second case is after the collision.
After the collision, the car's momentum is zero. But if the impact is too great, one of the pieces of the car may be thrown around.
I thought that the velocity of this piece thrown after the collision could be determined by its range.
I thought that when the velocity of this piece was determined, I would use the law of momentum conservation for this car and the piece attached to it.
The calculation is as stated above in the image below.
But I doubt its correctness. Please guide me in this question.
thanks

 

[kuruman]

You are supposed to answer the question assuming that you survey the collision site and deduce the speed of the car from which a piece thrown, presumably in the forward direction. Your equation has too many unknowns, namely the masses of the cars and the projection angle $\alpha$ all of which you don't know. Furthermore, the equation for the range that you used assumes that the projectile returns to the same height from which it was launched. This is not case here because found on the ground must have been launched above ground height. A reasonable assumption is that the piece started from some height $h$ above ground with intial horizontal velocity $V$ equal to the velocity of the car and landed at distance $R$ from the car. Can you find $V$ from estimates of $R$ and $h$?

 

[peace]

You are supposed to answer the question assuming that you survey the collision site and deduce the speed of the car from which a piece thrown, presumably in the forward direction. Your equation has too many unknowns, namely the masses of the cars and the projection angle $\alpha$ all of which you don't know. Furthermore, the equation for the range that you used assumes that the projectile returns to the same height from which it was launched. This is not case here because found on the ground must have been launched above ground height. A reasonable assumption is that the piece started from some height $h$ above ground with intial horizontal velocity $V$ equal to the velocity of the car and landed at distance $R$ from the car. Can you find $V$ from estimates of $R$ and $h$?

The reason the masses are unknown is that it is an analytical question. Of course, at the end we are asked to give a numerical example. I considered the angle myself and the question did not say anything about it. Because I wanted to use the range formula, I had to consider the angle as well.
I do not think the range has anything to do with height. Because the range is the maximum horizontal distance of the object. On the other hand, the coordinate system can be placed not on the ground but at the point where the piece is thrown.

 

[kuruman]

You need to derive a new range formula. The one you propose to use is valid only if the object lands at the same level from which it was launched. You can place your origin wherever you want, but the fact remains that any piece of the car that breaks off starts above ground and lands on the ground.

The range has a lot to do with height. Put a book on a table and give it a push so that it slides off and hits the floor. Once in the air, its horizontal velocity is constant. The longer it stays in the air, the farther away it hits the floor. The higher the table, the longer it stays in the air hence the longer the range.

 

[peace]

You need to derive a new range formula. The one you propose to use is valid only if the object lands at the same level from which it was launched. You can place your origin wherever you want, but the fact remains that any piece of the car that breaks off starts above ground and lands on the ground.

The range has a lot to do with height. Put a book on a table and give it a push so that it slides off and hits the floor. Once in the air, its horizontal velocity is constant. The longer it stays in the air, the farther away it hits the floor. The higher the table, the longer it stays in the air hence the longer the range.

You are right. I realized. I'm trying to find a new formula. Although I do not know how, now. But I try. Thanks for your guidance.

 

[kuruman]

You are right. I realized. I'm trying to find a new formula. Although I do not know how, now. But I try. Thanks for your guidance.

If you assume that the initial velocity of the piece is horizontal, the new formula is easy to find. The initial horizontal velocity is the velocity of the car and the initial vertical velocity is zero. This means that the time of flight is the same as that of an object released from rest at height $h$ above ground.

 

[peace]

If you assume that the initial velocity of the piece is horizontal, the new formula is easy to find. The initial horizontal velocity is the velocity of the car and the initial vertical velocity is zero. This means that the time of flight is the same as that of an object released from rest at height $h$ above ground.

Why is the initial horizontal velocity of the piece the same as the velocity of the car?
Does this mean that we have a free fall problem if the initial vertical velocity is zero?

 

[kuruman]

Why is the initial horizontal velocity of the piece the same as the velocity of the car?

Initially the piece is at rest relative to the car, therefore it has the speed of the car relative to the ground. The assumption is that, when the collision occurs, the car stops (almost) instantly while the piece breaks off and keeps on going at the speed of the car.

Does this mean that we have a free fall problem if the initial vertical velocity is zero?

That's what it means. The vertical motion is independent of the horizontal motion. You may have seen the demo of this. If not, take a look at this Shoot n' Drop.

 

[peace]

Initially the piece is at rest relative to the car, therefore it has the speed of the car relative to the ground. The assumption is that, when the collision occurs, the car stops (almost) instantly while the piece breaks off and keeps on going at the speed of the car.

My goal in the range formula was to find the velocity of the piece. So when the velocity of the piece is the same as the velocity of the car, what is the need for a range! Is the idea of using the law of conservation in the way I said above correct?

 

[kuruman]

So when the velocity of the piece is the same as the velocity of the car, what is the need for a range!

Because using the range you are supposed to deduce the speed. You don't know if the speed of the car and the piece was 30 mph, 45 mph or whatever when the piece broke off. You are supposed to estimate that speed from knowing how far from the car the piece landed which is a distance that you can measure with a tape or just estimate by looking at it.

 

[haruspex]

Why is the initial horizontal velocity of the piece the same as the velocity of the car?

It won't be exactly that. It will have taken some momentum to wrench the piece off the car, but the calculated horizontal speed just after collision for a piece will be a lower bound for the speed of the car just before collision.

 

[peace]

Because using the range you are supposed to deduce the speed. You don't know if the speed of the car and the piece was 30 mph, 45 mph or whatever when the piece broke off. You are supposed to estimate that speed from knowing how far from the car the piece landed which is a distance that you can measure with a tape or just estimate by looking at it.

I realized.
So in conclusion: I have to find a relationship between the range and the height of the piece and its velocity. And then estimate the initial velocity of the car using the law of conservation.

 

[kuruman]

I realized.
So in conclusion: I have to find a relationship between the range and the height of the piece and its velocity. And then estimate the initial velocity of the car using the law of conservation.

The law of momentum conservation does not come into the picture. If you have "a relationship between the range and the height of the piece and its velocity", all you need to do is estimate the initial height and the range and voilà, you have the velocity. Don't forget that this is a lower bound as @haruspex indicated in post #11.

 

[peace]

The law of momentum conservation does not come into the picture. If you have "a relationship between the range and the height of the piece and its velocity", all you need to do is estimate the initial height and the range and voilà, you have the velocity. Don't forget that this is a lower bound as @haruspex indicated in post #11.

I found the desired relationship. For the initial velocity of the piece.
But I still do not know what to do with this relationship.
How do I estimate the initial velocity of the car?

 

[haruspex]

I found the desired relationship. For the initial velocity of the piece.
But I still do not know what to do with this relationship.
How do I estimate the initial velocity of the car?

I don't know how to make it any clearer than what I wrote in post #11.

 

[kuruman]

I found the desired relationship. For the initial velocity of the piece.
But I still do not know what to do with this relationship.
How do I estimate the initial velocity of the car?

I am beginning to suspect that you are looking for a precise answer, something like 23.596 m/s. That is not the case here. How about something like "The speed of the car cannot have been higher than _____ m/s"? Can you do that with the equation you derived? If not, please post your equation and try to explain exactly what it is that prevents you from using it. We cannot read your mind.

 

[peace]

I am beginning to suspect that you are looking for a precise answer, something like 23.596 m/s. That is not the case here. How about something like "The speed of the car cannot have been higher than _____ m/s"? Can you do that with the equation you derived? If not, please post your equation and try to explain exactly what it is that prevents you from using it. We cannot read your mind.

Do you mean to consider a range for the initial velocity of the car?
But the only formula available is the piece velocity formula after the collision. I do not think that way we can consider a range for the initial velocity of the car. I do not know.

 

[kuruman]

How do you think the velocity of the piece at the moment it comes loose is related to the velocity of the car has at that moment? Assume that the car is moving at 20 m/s. The velocity of the piece at that moment is most likely
A. 2 m/s
B. 15 m/s
C. 35 m/s
D. 3×10⁸ m/s

What do you think and why?

 

[peace]

How do you think the velocity of the piece at the moment it comes loose is related to the velocity of the car has at that moment? Assume that the car is moving at 20 m/s. The velocity of the piece at that moment is most likely
A. 2 m/s
B. 15 m/s
C. 35 m/s
D. 3×10⁸ m/s

What do you think and why?

I think just having the velocity of a car is not enough. But I feel the velocity of the piece is lower than the velocity of the car.

 

[kuruman]

I think just having the velocity of a car is not enough.

Why not? What is your thinking process that says it cannot be so?

But I feel the velocity of the piece is lower than the velocity of the car.

This is not about feelings. What is your reasoning? Which of the 4 choices I gave is most likely and why?

 

[peace]

Why not? What is your thinking process that says it cannot be so?

This is not about feelings. What is your reasoning? Which of the 4 choices I gave is most likely and why?

Yes, you are right. But I really do not know. I can not find a physical reason.

 

[kuruman]

Can you perhaps eliminate some of the 4 choices? If so, on what grounds? Consider them one at a time starting with A.

 

[peace]

Can you perhaps eliminate some of the 4 choices? If so, on what grounds? Consider them one at a time starting with A.

The piece has zero velocity relative to the car. But it has a velocity relative to the ground. A velocity equal to the velocity of a car. s I think there should not be much difference between the velocity of the piece at the moment of loosening and the velocity of the car. That's why I think the last option should be removed.

 

[kuruman]

That's good thinking about option D which is the speed of light. Cars are not known to go that fast. Can you apply similar reasoning to additional option(s)?

 

[peace]

That's good thinking about option D which is the speed of light. Cars are not known to go that fast. Can you apply similar reasoning to additional option(s)?

I do not think that the velocity of the piece is faster than the velocity of the car. When two cars collide, a lot of energy is released. the energy is spent separating this piece from the car. Because this piece was part of the car before the collision. For example, it was part of the car body. And maybe there is not enough energy to throw this piece at high velocity. That's why I don't think the velocity of this piece when it came off was faster than the velocity of the car. That's why I think the third option should be removed.

 

[kuruman]

So far so good. There are two options left both of which are lower than the known speed of the car. Any ideas about eliminating either one of them?

 

[peace]

So far so good. There are two options left both of which are lower than the known speed of the car. Any ideas about eliminating either one of them?

My first reason was that they should not be too different. Because they both have the same velocity relative to the ground before the cars collide. But the car velocity immediately goes to zero after the collision. The piece also slows down immediately. Because its velocity was the same as the car. But as I said in the previous reason, some of the energy released in this collision is spent on this piece. Of course, as I said, I do not think that this energy is so much that it can accelerate the piece a lot. I think the most appropriate velocity is the first option.

 

[kuruman]

My first reason was that they should not be too different.

What are "they" that should not be different? The speeds of the piece and car or the remaining choices (A) and (B)?

Because they both have the same velocity relative to the ground before the cars collide. But the car velocity immediately goes to zero after the collision. The piece also slows down immediately.

What do you think slows the piece down? Does whatever that is act on the piece as it flies through the air?

Because its velocity was the same as the car. But as I said in the previous reason, some of the energy released in this collision is spent on this piece. Of course, as I said, I do not think that this energy is so much that it can accelerate the piece a lot. I think the most appropriate velocity is the first option.

Suppose you said to an experienced accident investigator that the speed of the car before the collision was 2 m/s and he said to you, "The evidence on the ground does not support this", what do you think he saw?

 

[peace]

What are "they" that should not be different? The speeds of the piece and car or the remaining choices (A) and (B)?

The speeds of the piece and car. Of course I'm not sure.

What do you think slows the piece down? Does whatever that is act on the piece as it flies through the air?

I do not know the reason. Because the car stops after the collision and on the other hand the piece is with it, so I thought the speed of the piece would also decrease.

Suppose you said to an experienced accident investigator that the speed of the car before the collision was 2 m/s and he said to you, "The evidence on the ground does not support this", what do you think he saw?

I think he saw pieces of the car body scattered over a long distance. In fact, he saw that the range of those pieces was so high that the collision was most likely severe, so the speed of the car should not be that number.

 

[kuruman]

The speeds of the piece and car. Of course I'm not sure.

How can you not be sure that you meant the speeds of the piece and the car? You know what you meant, I don't that's why I asked.

I do not know the reason. Because the car stops after the collision and on the other hand the piece is with it, so I thought the speed of the piece would also decrease.

"I do not know the reason" is not sufficient explanation when you claim something to be true. Consider a possible specific piece, a baby seat (without a baby) next to the driver's seat in a convertible with the top down. When the car collides and stops, the baby seat will stay in place or fly off depending on whether it is strapped in or not. Just because the car stops does not mean that the baby seat will have to stop. There is a reason why it stops. The fact that the baby seat was found on the ground farther away than the car indicates that it did not stop like the car but flew through the air and was stopped by the ground. That's the motion that we want to analyze, namely reconstruct the motion of the baby seat through the air.

I think he saw pieces of the car body scattered over a long distance. In fact, he saw that the range of those pieces was so high that the collision was most likely severe, so the speed of the car should not be that number.

The speed of the car should not be what number? You did not mention any numbers. It seems that you are linking the position of the pieces with the "severity" of the collision. How does one measure this severity? Did you really mean to say "the speed of the car before the collision must have been very high for the baby seat to land where it did"?