This math question is killing me! solution needed asap!!!!?

Question

two runners, lisa and alice, are leading the field in a long-distance race. they are both running at 5m/s, with lisa 10m behind alice. when alice is 50m from the tape, lisa accelerates but alice doesn't. what is the least acceleration lisa must produce to overtake alice?
if instead alice accelerates at 0.1 m/s^2 up to the tape, what is the least acceleration lisa must produce?

answers:0.2/s^2; 0.34 m/s^2

Answer 1

the time taken by alice for covering 50 m @ 5ms^-1 is 10 secs
in 10 secs lisa has to cover 60 m with an initial vel of 5 ms^-1 and acceleration of say 'a' ms^-2.the equation is
60=5(10)+1/2a(100)
50a=10and so a=10/50=0.2 m/s^2
if alice accelerates @ 0.1m/s^2 if time taken by alice is 't' the equation is 50=5t+1/2*0.1*t^2
=>5t^2+500t-5000=0
=>t^2+100t-1000=0
solve for t and substitute in 60=5t+1/2at^2 to get 'a'

Answer 2

Both initially running at 5 m/s

L is 10m futher away from the tape than A
A is 50m from the tape

(1)
A will cross 50m in 10 s (slow girl)
L needs to cross 60m in 10 s and taking into account a changing velocity must produce an average velocity of > 6m/s

She will therefore need to change her velocity from 5m/s to >7m/s over this last 10 s

A = Δv/t

A > 2/10 m/s^2
A > 0.2 m/s^2

(Note, if L accelerates at 0.2 m/s^2 she will wind up in a tie with A)

(2)
In this second scenario, A accelerates also at 0.1 m/s^2 over the last 50m.

Alice's initial speed is 5 m/s
Alice's speed over the last 50m can be represented by
Sa = 5+0.1t m/s where t is the time in sec from the point of acceleration towards the finish.

How long does it take Alice to get to the finish (50m away)?
D = S x t (distance = speed x time)

50 = (5 + 0.1t) t
t^2/10 + 5t - 50 = 0 which is a quadratic of the form ax^2+bx+c-0 so we can apply the quadratic equation to determine the roots. Maybe it's better if I write the quadratic as 0.1t^2 + 5t - 50 = 0

t = (-5 +/- SQRT(25+4 x 50 x 0.1))/0.2
= (-5 +/- SQRT(45)) x 5
NOTE - only (-5 + SQRT(45)) x 5 makes sense as an answer here - no negative time for Alice!

approx = 8.541 sec

L needs to cross 60m in < 8.541 s

Her average speed must now exceed 7.025 m/s and so she will need to have a final speed of 9.05m/s

So her acceleration is (9.05-5.00)/8.541 = 0.47 m/s^2

Answer 3

The long but easy solution to this problem is in five parts. The first problem is in two parts and the second problem is in three parts.

Find the time it will take Alice to finish the race. This is easy; at 5m/s it will take 10s to run 50m. So, Lisa has to cover her 60m is less than ten seconds. We can solve this by letting Lisa win by one second or one tenth of a second, we can also solve this by letting Lisa win by one meter or one tenth or one hundredth of a meter.

I decided to solve this by letting Lisa win this by one centimeter. This will make it a little more exciting! But, I will show you in the solution where you can make changes to let Lisa win by 1/00 of a second if you want!

The equation we can use to let Lisa win by 1cm is this: d = initial velocity x time + ½ acceleration times time squared. (d = vit + ½ at^2) We need to solve this equation for acceleration. We get: a = 2(d – vit) / t^2. ('vit' means 'initial velocity times time)

To win by 1cm I put 60.01m for‘d’. To let Lisa win by 1/100 of a second put 60m in for‘d’ and 9.99s in for the time.

For the second problem we have to first find how long it will take Alice to cover the 50m while accelerating at 0.1m/s^2. We can either use the quadratic equation or use two different equations. It would be easier on this computer to use two equations to find out how long it will take Alice to finish the race if at 50m she accelerates 0.1m/s^2.

We will use this equation to find the time: a = vf – vi / t. Solved for ‘t’ we get t = vf – vi / a. ('vf' means final velocity and 'vi' means initial velocity.) As you can see we need to find the final velocity in order to use this equation. But, this is not problem because we know the equation to use to find this.

To find the final velocity we will use vf^2 = vi^2 + 2ad. This gives us a final velocity of 5.9m/s. Put this into the equation above and we find that Alice will finish the race in 9s.

Now, we use the same equation we used before to find Lisa’s acceleration. Again, you can use either 60.01m to let Lisa win by 1cm or you can use 60m and 8.99s to let Lisa win by 1/100s.

This is really fun if you know what you are doing!