100 meter running with constant acceleration

[MariusM]

Hello, I am currently by some reason struggling with this problem. I cannot seem to get the correct reasoning here. I would appreciate any help!

1. Homework Statement
Ann and Lisa cross the 100 m finish line at the same time, 10.2 s after start.
Both had constant acceleration until they reached max speed.
Ann used 2.0 s to reach max speed.
Lisa used 3.0 s to reach max speed.
The max speed was sustained until they reached the finish line.

1) What was their respective acceleration?
2) What was their respective max speed?
3) Who was in the lead after 6.0 s and by how many meters?

Homework Equations

[/B]
$v=v_0+at$
$x=x_0+v_0t+.5at^2$
$v^2=v^2_0+2aΔx$

The Attempt at a Solution

[/B]
0 = start, 1 = max speed reached, 2 = goal.

For Ann:
$t_1=2s$, $t_2=8.2s$, $v_0=0$, $x_0=0$, $v_1=v_2$

$v_1=at_1$, $x_1=.5at^2_1$, $v_1t_2=x_2$

Here I am struggling, If I do this for both I get 8 unknowns with 6 equations.

 

[ehild]

1. Homework Statement
Ann and Lisa cross the 100 m finish line at the same time, 10.2 s after start.
Both had constant acceleration until they reached max speed.
Ann used 2.0 s to reach max speed.
Lisa used 3.0 s to reach max speed.
The max speed was sustained until they reached the finish line.

The Attempt at a Solution

[/B]
0 = start, 1 = max speed reached, 2 = goal.

For Ann:
$t_1=2s$, $t_2=8.2s$, $v_0=0$, $x_0=0$, $v_1=v_2$

$v_1=at_1$, $x_1=.5at^2_1$, $v_1t_2=x_2$

Here I am struggling, If I do this for both I get 8 unknowns with 6 equations.

You know that Ann used t1=2 s to reach the maximum speed. You also know that the whole time is 10.2 s and the whole distance is 100 m. express v1 and x1 in terms of Ann's acceleration, a. Try to find a from the data given.
Do the same for Lisa.

 

[MariusM]

Got it. Thank you! I was to narrow with my thinking, "dumbing" the problem down too much and didn't consider it would have to involve more thinking than just the 3 equations i gave. The fourth I consider was just some rubbish once I checked the dimensions. $v_1t_1+x_1=100$ was the last one necessary and I was OK. Again, thanks!

 

[ehild]

You are welcome :)

 

[HalfLostAndSearching]

Hi there! It looks like you're on the right track with using the equations for constant acceleration to solve this problem. Let's go through each of the parts and see if we can come up with a solution together.

1) To find the acceleration, we can use the equation v=v_0+at, where v is the final velocity, v_0 is the initial velocity, a is the acceleration, and t is the time. We know that Ann reaches her max speed in 2 seconds, so we can plug in those values:

v = 0 + a(2)
v = 2a

We also know that her final velocity at the finish line is the same as her max speed, so we can set that equal to v_2:

v_2 = 2a

Now, we can do the same thing for Lisa, but using a different time value of 3 seconds:

v = 0 + a(3)
v = 3a
v_2 = 3a

2) To find the max speed, we can use the equation v^2=v^2_0+2aΔx, where v is the final velocity, v_0 is the initial velocity, a is the acceleration, and Δx is the change in position. We know that Δx is 100 meters, and we can use the values we found for v_2 in the previous step to solve for the max speeds:

v^2 = 0 + 2a(100)
v^2 = 200a
v = √(200a)

For Ann:

v = √(200a) = 2a

For Lisa:

v = √(200a) = 3a

3) To find who is in the lead after 6 seconds, we can use the equation x=x_0+v_0t+.5at^2, where x is the final position, x_0 is the initial position, v_0 is the initial velocity, a is the acceleration, and t is the time. We know that after 6 seconds, Ann and Lisa have both reached their max speeds and are traveling at a constant velocity. So, we can use the equation v=v_0+at to find their respective positions after 6 seconds.

For Ann:

v = 2a (from earlier)
v = v_0 + at
2a = v_