Calculating Distance and Acceleration in Circular Motion of Two Cyclists

[jjiimmyy101]

Two cyclists A and B, are traveling counterclockwise around a circular track at a constant speed of 8 ft/sec at the instant shown. If the speed of A is increased at aA = SA ft/sec^2, where SA is in ft, determine the distance measured counterclockwise along the track from B to A between the cyclists when time is = 1 sec. What is the magnitude of the acceleration of each cyclist at the instant?

To find the length of an arc, you use the equation arc=theta*radius, but how do you encorporate the time into this? I don't know how far cyclist A moves.

To find the magnitude of acceleration you can use the sqrt of a(normal)^2 + a(tangential)^2.

aB =1.28 ft/sec^2

because a(tangential)=0 (constant velocity) and a(normal)=64/50

I don't know what aA is equal to.

Any suggestions?

I posted a picture too.

 

[Doc Al]

Originally posted by jjiimmyy101
To find the length of an arc, you use the equation arc=theta*radius, but how do you encorporate the time into this? I don't know how far cyclist A moves.

First find the distance between A & B at time t=0 (the instant shown). Then find out how far each moves in the next second. If it wasn't for that fact that A is accelerating, they would move the same distance, thus maintaining the same separation. But A gains some distance over B: ΔX = 1/2at².

Also, A gains some speed: ΔV = at.

 

[jjiimmyy101]

at time t=0 the distance between them is 104.72ft

B moves 8ft in the next second because it is constant

but i still don't get how far A moves.

and how do I use deltaX and deltaV

 

[Doc Al]

You need to understand the basic formulas for uniform accelerated motion. One key relation is $d = v_0t + \frac{1}{2}at^2$, which describes the distance traveled in time t. ($v_0$ is the initial speed.) Another useful formula gives the speed after time t: $v = v_0 + at$. You will need both of these to understand how "A" moves.

B is just moving at a constant speed (tangential acceleration = 0). I believe you understand that, but note that the above equations apply if you set a=0. (Please try this!)

Of course, the above only applies to the tangential motion. To find the full acceleration, you must add the centripetal acceleration.

Note: Δ just means "change"; ΔX means change in x.

 

[jjiimmyy101]

Thanks!

I got the final answers.

Thank-you.