# Linear eqns: Driver 1 goes V_1 for t_1 hrs; Driver 2 goes V_2 for another t_2 hours.

#### Aion

##### New member
1.34: A motorist starts and drives at a constant speed $$\displaystyle V_1\, \mbox{km/h}.$$ After $$\displaystyle t_1$$ hours, another driver starts from the same location and drives along the same road at constant speed $$\displaystyle V_2\, \mbox{km/h},$$ where $$\displaystyle V_2\, >\, V_1$$ and, after another $$\displaystyle t_2$$ hopurs, catches up with the first driver.

(a) Derive a formula for calculating $$\displaystyle t_2$$ when $$\displaystyle V_1,\, V_2,$$ and $$\displaystyle t_1$$ are known.

Thanks for the help everyone. I finally understand how to solve this. I think I found this problem hard since I wasn't familiar with the formula:

D = rΔt

Solution:

V1(t1+t2) = V2(t2)

t2=(V1t1)/(V2-V1)

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#### Aion

##### New member
Linear equation problem

#### Subhotosh Khan

##### Super Moderator
Staff member
1.34: A motorist starts and drives at a constant speed $$\displaystyle V_1\, \mbox{km/h}.$$ After $$\displaystyle t_1$$ hours, another driver starts from the same location and drives along the same road at constant speed $$\displaystyle V_2\, \mbox{km/h},$$ where $$\displaystyle V_2\, >\, V_1$$ and, after another $$\displaystyle t_2$$ hopurs, catches up with the first driver.

(a) Derive a formula for calculating $$\displaystyle t_2$$ when $$\displaystyle V_1,\, V_2,$$ and $$\displaystyle t_1$$ are known.

I believe there exist two functions f(t), and h(t) such that f(t1+t2) = h(t2), but i'm still uncertain how to solve this problem.

Any insights would be appreciated!
What are your thoughts regarding the assignment?

Please share your work with us ...even if you know it is wrong.

If you are stuck at the beginning tell us and we'll start with the definitions.

http://www.freemathhelp.com/for

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#### Dr.Peterson

##### Elite Member
I believe there exist two functions f(t), and h(t) such that f(t1+t2) = h(t2), but i'm still uncertain how to solve this problem.Any insights would be appreciated!
I think you're making it more complicated than it is. There is no need for unknown functions.

Can you write an expression for the position of the first driver at time t (hours after the first driver starts)?

Can you write another expression for the position of the second driver at time t?

Then you can write an equation, stating that they are at the same position at time t1 + t2.

#### HallsofIvy

##### Elite Member
"A motorist starts and drives at constant speed V1 km/h"
So after t hours he has gone V1t km.

"After t1 hours another motorist starts from the same location and drives along the same road at V2 km/h, where V2> V1, and after another t2 hours catches up with the first motorist."
So t hours after the first motorist starts, the second motorist will have gone V2(t- t1) km. He will have caught up with the first motorist when they have both gone the same distance. What equation represents this same distance? If you solve this equation for t_2 in terms of the other variables, what do you get?

Another way of looking at this is that the second motorist's speed, relative to the first motorist is V2- V1. When the second motorist starts, the first motorist is distance V1t1 away. The second motorist must cover distance V1t1 at speed V2- V1 which will take... how many hours?

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#### Aion

##### New member
Sorry but I don't comprehend any of your answers. Also, none of you has solved the problem which I clearly stated above.

$$\displaystyle t_2$$ = $$\displaystyle V_1(t_1)/(V_2-V_1),$$

Was hoping someone might show me their solution since I've already attempted to solve this for many hours at end!

#### Aion

##### New member
So if I say that:

$$\displaystyle V_1(t_1+t_2)=V_2t_2$$

$$\displaystyle V_1t_1+V_1t_2-V_2t_2 = 0$$

$$\displaystyle = t_2(V_1-V_2) = -V_1t_1$$

$$\displaystyle = (-V_1t_1)/(V_1-V_2) <==> ((-1)(V_1t_1))/((-1)(-V_1+V_2)) = t_2$$

$$\displaystyle = (V_1t_1)/(V_2-V_1) = t_2$$

Thanks for the help, although I would have appreciated had you used the right variables for the given problem.