Kinematics question -- 2 runners racing with different acceleration rates

[heythere1010]

Homework Statement

Justin is racing Mark. Justin accelerates from rest at the starting point at 1.2 m/s^2 [E]. Mark is still adjusting his equipment and 2.0 s later begins to accelerate at 1.5 m/s^2 [E]. Where and when does Mark catch up to Justin?

Homework Equations

The Attempt at a Solution

I think I need to create two formulas, set them equal to each other, and use the quadratic formula. I'm not sure how to create the formulas though.

 

[haruspex]

Homework Statement

Justin is racing Mark. Justin accelerates from rest at the starting point at 1.2 m/s^2 [E]. Mark is still adjusting his equipment and 2.0 s later begins to accelerate at 1.5 m/s^2 [E]. Where and when does Mark catch up to Justin?

Homework Equations

The Attempt at a Solution

I think I need to create two formulas, set them equal to each other, and use the quadratic formula. I'm not sure how to create the formulas though.

What equations of motion under constant acceleration do you know? There are a standard five "SUVAT" equations.

 

[heythere1010]

What equations of motion under constant acceleration do you know? There are a standard five "SUVAT" equations.

v=u+at, d=ut+1/2at^2, d=vt-1/2at^2

 

[haruspex]

v=u+at, d=ut+1/2at^2, d=vt-1/2at^2

Ok. You can see that there are five variables that recur, two of speed, one acceleration, one distance, and one time. Which of those five is of no interest here? Find the equation that does not involve that variable.

 

[heythere1010]

Ok. You can see that there are five variables that recur, two of speed, one acceleration, one distance, and one time. Which of those five is of no interest here? Find the equation that does not involve that variable.

So d=ut+1/2at^2.

 

[haruspex]

So d=ut+1/2at^2.

Right. So write out that equation for each runner, and the relationships between the times and distances in the two equations.

 

[heythere1010]

Right. So write out that equation for each runner, and the relationships between the times and distances in the two equations.

How would I set them up if there are two unknown variables t and d?

 

[haruspex]

How would I set them up if there are two unknown variables t and d?

There'll be two unknowns in each equation, but in each case there's a known relationship between the unknown in one equation and the corresponding unknown in the other equation. So you have enough equations to solve for all unknowns.

 

[heythere1010]

There'll be two unknowns in each equation, but in each case there's a known relationship between the unknown in one equation and the corresponding unknown in the other equation. So you have enough equations to solve for all unknowns.

Which variables are related?

 

[haruspex]

Which variables are related?

The accelerations are known to be different, and the initial velocities are both known. What does that leave?

 

[heythere1010]

The accelerations are known to be different, and the initial velocities are both known. What does that leave?

Distance. But I don't think the equation would be d=0.6t^2 and d=0.75t^2 right?

 

[haruspex]

Distance. But I don't think the equation would be d=0.6t^2 and d=0.75t^2 right?

That's right, because the the times are different. But they are related - how?

 

[heythere1010]

That's right, because the the times are different. But they are related - how?

The distance is the same.

 

[haruspex]

The distance is the same.

Sure, but how are the two times related? You are told.

 

[heythere1010]

Sure, but how are the two times related? You are told.

Yes, one is two seconds longer, t+2.

 

[haruspex]

Yes, one is two seconds longer, t+2.

Right. So what equations do you have now?

 

[heythere1010]

Right. So what equations do you have now?

I don't know where t+2 would go.

 

[haruspex]

I don't know where t+2 would go.

Simplest way to think about it is this: let t₁ be the first runner's acceleration time and t₂ be the second runner's acceleration time. That allows you to write the two SUVAT equations. Now you just have to relate t₁ and t₂ by the fact that they differ by 2 - the tricky part is to make sure you get it the right way round. Think: which runner runs for longer?

 

[heythere1010]

Simplest way to think about it is this: let t₁ be the first runner's acceleration time and t₂ be the second runner's acceleration time. That allows you to write the two SUVAT equations. Now you just have to relate t₁ and t₂ by the fact that they differ by 2 - the tricky part is to make sure you get it the right way round. Think: which runner runs for longer?

I'm still not sure how I add the 2 seconds into the equation.

 

[haruspex]

I'm still not sure how I add the 2 seconds into the equation.

If Justin (first runner) runs for t₁ seconds, how long does Mark run for?

 

[heythere1010]

If Justin (first runner) runs for t₁ seconds, how long does Mark run for?

t-2.

 

[haruspex]

t-2.

Right, so plug t-2 in place of t in Mark's equation.

 

[heythere1010]

Right, so plug t-2 in place of t in Mark's equation.

Yes, I did this, but when I make it equal to the other equation, both my answers are positive.

 

[haruspex]

Yes, I did this, but when I make it equal to the other equation, both my answers are positive.

Please post your working.
It is quite possible they are both positive, but is one less than 2 seconds?

 

[heythere1010]

Please post your working.
It is quite possible they are both positive, but is one less than 2 seconds?

0.75(t-2)^2=0.6t^2
Solving using the quadratic formula gave me 18.9 and 1.1.

 

[haruspex]

0.75(t-2)^2=0.6t^2
Solving using the quadratic formula gave me 18.9 and 1.1.

OK, but with Mark running for only t-2 seconds, what do you think t=1.1 means?

 

[heythere1010]

OK, but with Mark running for only t-2 seconds, what do you think t=1.1 means?

Right haha. Really late where I am. Thanks for the help!