Acceleration, Velocity, Position and Time

[PequodLZ]

There is a race between two contestants on who can can reach 50m first.
For Contestant A we are given an acceleration equation as a=e^v where v is velocity
For Contestant B we are given a position equation as x=a^-4 where a is acceleration
Who would win the race?

Im normally used to these equations having time on them wherein I would just derive or integrate to find the other equations.

 

[Deleted member 4993]

There is a race between two contestants on who can can reach 50m first.
For Contestant A we are given an acceleration equation as a=e^v where v is velocity
For Contestant B we are given a position equation as x=a^-4 where a is acceleration
Who would win the race?

Im normally used to these equations having time on them wherein I would just derive or integrate to find the other equations.

You wrote:

For Contestant A we are given an acceleration equation as a=e^v where v is velocity → \(\displaystyle \frac{dv}{e^v} = dt .\)→ now integrate....

Please show us what you have tried and exactly where you are stuck.

Please follow the rules of posting in this forum, as enunciated at:

READ BEFORE POSTING​

​

Please share your work/thoughts about this problem

 

[nasi112]

Absolutely B. Can you see why?

 

[PequodLZ]

Sorry got very busy as of late.
I feel as if I'm wrong but:
For contestant A we are given a=e^v
I know that a=vdv/dx
I transpose a and ds to get dx=vdv/a
I get the definite integral of dx from 0 to 50 since 50 is the end of the race.
I get the definite integral of vdv/a from 0 to v where v will be the final velocity when we hit 50m
So now we have 50=∫ (v/e^v)dv (from 0 to v) (plugging in a=e^v as well)
I get 50= -(v/e^v)-(1/e^v)+1
I got v=-3.133878896 (getting a negative here feels wrong but I cant think of another solution)
So when we hit the finish line x=50 his velocity is v=-3.133878896

Now I also know that a=dv/dt
I transpose a and dt to get dt=dv/a
I get the definite integral of dt from 0 to t where t will be the time we got to the finish line
I get the definite integral of dv/a from 0 to -3.133878896 since that will be the velocity we reach once we are at the finish line.
So now we have t= ∫ (1/e^v)dv (from -3.133878896 to 0)
t= 21.5s for Contestant A


For Contestant B we have x=1/a⁴ manipulating can also give a=(1/x)^1/4
I know that dx/v=dv/a manipulating can give adx=vdv
I get the definite integral of adx from 0 to 50 since 50 is the finish line.
I get the definite integral of vdv from 0 to v giving me the velocity when x=50.
So now we have ∫((1/x)^1/4))dx (from 0 to 50) = ∫(v)dv (from 0 to v)
This gives us 25.07=v²/2
So when x=50, v=12.535^1/2
Lastly we all know v=dx/dt
So 12.535^1/2=(50-0)/(t-0)
So t=14.12s for Contestant B

So contestant B wins... I think...

 

[PequodLZ]

Absolutely B. Can you see why?

I did, but only after a lengthy solution. If you have some sort of neat trick please enlighten me.