Classical mechanics : constant acceleration

[REVIANNA]

Homework Statement

Two cyclists, Anant and Beth, are standing on a straight road a distance d=22.0 m apart. Simultaneously, they begin to move toward each other, Anant with acceleration a_A=0.2 m/s^2 and Beth with acceleration a_b=0.10 m/s^2.

Just as they begin to move, a bee, sitting on Anant’s helmet, begins to fly toward Beth at a constant speed v=11.0m/s. Once the bee reaches Beth, it instantly turns around and flies back toward Anant. As the cyclists move toward each other, the bee continues to fly back-and-forth between the two until they pass each other. The speed of the bee remains constant.How long does it take for the riders to pass each other?

What is the distance covered by the bee between the start and the moment Anant and Beth pass each other?

What is the average velocity of the bee between the start and the moment Anant and Beth pass each other?[/B]

Homework Equations

x_0 for anant =0
x_0 for beth=22

The Attempt at a Solution

I found out the time it takes for them to cross each other.
0.5*0.2*t^2=22-(0.5*0.1*t^2)
therefore t=12.11060 seconds

I correctly found the distance covered by the bee
time* constant speed=12.11060*11=133.2166 m

but I trouble finding AVERAGE VELOCITY because I don't how to think about the DISPLACEMENT of the bee.

 

[Andrew Mason]

The Attempt at a Solution

I found out the time it takes for them to cross each other.
0.5*0.2*t^2=22-(0.5*0.1*t^2)
therefore t=12.11060 seconds

I correctly found the distance covered by the bee
time* constant speed=12.11060*11=133.2166 m

but I trouble finding AVERAGE VELOCITY because I don't how to think about the DISPLACEMENT of the bee.

Welcome to PF Revianna!

Average velocity is just the total displacement divided by total time. So you have to find the location of the point at which the cyclists cross. The displacement of that point relative to the point where the fly started its journey is the total displacement.

AM

 

[Simanto Rahman]

Welcome to PF Revianna!

Average velocity is just the total displacement divided by total time. So you have to find the location of the point at which the cyclists cross. The displacement of that point relative to the point where the fly started its journey is the total displacement.

AM

The trouble isn't finding the average velocity. It will be zero considering in directional aspect, it goes no where. So its direction vector will be zero. But how many times the bee goes back and forth is the real trouble. Because every second the distance between them decreases so bee travels less distance and spends less time to reach one person from the other. I see no way but to cross reference it. A multi-valued chart would be easy to find that bit I think

 

[Andrew Mason]

The trouble isn't finding the average velocity. It will be zero considering in directional aspect, it goes no where. So its direction vector will be zero. But how many times the bee goes back and forth is the real trouble. Because every second the distance between them decreases so bee travels less distance and spends less time to reach one person from the other. I see no way but to cross reference it. A multi-valued chart would be easy to find that bit I think

The fly ends up traveling from Point A (the position of Anant's helmet at time 0) to Point B (the position of Anant's helmet 12.11 seconds later). So you cannot say that its velocity is zero. Its average velocity is that displacement divided by 12.11 seconds, which is clearly not zero.

AM

 

[Simanto Rahman]

So you cannot say that its velocity is zero.

I see your point. I guess I put in the Work aspect instead of velocity aspect. But I don't think I was far off about the distance problem. It is going back and forth and every time traveling less and less distance for lesser time. So the whole calculation would require individual calculation of the distance the bee travels each time it reaches from one person to the other.

 

[REVIANNA]

@Andrew Mason
your suggestion to : calculate the position where the cyclists cross (by substituting the time found in the displacement equation) and the displacement of that point relative to the point where the fly started i.e x=0 its journey is the total displacement WORKS.

but how do you know that the BACK AND FORTH motion of the fly exactly equals the displacement of the cyclists.

 

[REVIANNA]

The fly ends up traveling from Point A (the position of Anant's helmet at time 0) to Point B (the position of Anant's helmet 12.11 seconds later). So you cannot say that its velocity is zero. Its average velocity is that displacement divided by 12.11 seconds, which is clearly not zero.

so we just have to think about the definition of displacement i.e final position (where they cross) - initial position x=0?

 

[Simanto Rahman]

A displacement is the shortest distance from the initial to the final position of a point P. Thus, it is the length of an imaginary straight path, typically distinct from the path actually traveled by P. A displacement vector represents the length and direction of this imaginary straight path.

 

[haruspex]

I see your point. I guess I put in the Work aspect instead of velocity aspect. But I don't think I was far off about the distance problem. It is going back and forth and every time traveling less and less distance for lesser time. So the whole calculation would require individual calculation of the distance the bee travels each time it reaches from one person to the other.

Revianna already found the scalar distance covered. It follows very easily from the first part of the question (the time taken for the two cyclists to meet).

 

[Andrew Mason]

so we just have to think about the definition of displacement i.e final position (where they cross) - initial position x=0?

The displacement from A to B is a vector drawn with tail at A and arrow head at B whose length represents the straight line distance between A and B in the inertial rest frame of the earth. In this case we are dealing with two dimensional vectors so the vector direction is one of two directions along the x-axis ie. either + or -.

but how do you know that the BACK AND FORTH motion of the fly exactly equals the displacement of the cyclists.

Because the question says that he starts at Anant's helmet when the two cyclists are 22 m apart and he goes back and forth between Anant and Beth until they cross paths (ie. until Anant and Beth have the same x coordinate position).

AM

 

[REVIANNA]

@Andrew Mason
I got it.Thanks.