Calculate Time Difference: Tim vs. Rick Running and Walking

[hieule]

Tim and Rick both can run at speed v_r and walk at speed v_w, with v_r greater than v_w. They set off together on a journey of distance D. Rick walks half of the distance and runs the other half. Tim walks half of the time and runs the other half.

the time it takes tim to cover the distance is:
t_R =((D/2)/v_w)+((D/2)/v_r)

the time it takes rick to cover the distance is:
t_T =2*D/(v_w+v_r)

In terms of given quantities, by what amount of time, Delta t, does Tim beat Rick?
It will help you check your answer if you simplify it algebraically and check the special case v_r = v_w.
Express the difference in time, Delta t in terms of v_r, v_w, and D.

this is an easy problem because all i need to do is take the Rick's time minus Tim's time right? I've done it and my answer is incorrect.

 

[Parth Dave]

You would take tim's time minus rick's time since tim takes less time. After that it is just algebra. All you have to do is simplify it.

 

[hieule]

You would take tim's time minus rick's time since tim takes less time. After that it is just algebra. All you have to do is simplify it.

that's my problem. i couldn't simplify it correctly. i simplified it to be

(2D(VwVr)-D/2(Vr+Vw)(Vw+Vr))/((Vw+Vr)(VwVr))
but this is not the answer

 

[dwade12]

A. use the formula d=v/t and rearrange it for time to t=d/v, since you have two v values (v_r and v_w), change the equation to t= (d/2vw)+(d/2vr). The two is in there because you multiple the distance by (1/2) which puts the two into the denominator.
B. The average speed is found by rearranging the answer to the above equation to answer for vw+vr, instead of time. You don't have to separate vw+vr when you get it on to the left side.
E. you DO have to subtract tims time from ricks because even though tim finishes first, rick takes LONGER thus a greater value for time. (or time would be negative otherwise)

 

[rwisz]

I would approach this problem by first defining all variables and their units. In this case, D would represent distance, v_r and v_w would represent speeds, and t_R and t_T would represent times. I would also clarify that the units for distance should be consistent with the units for speed and time.

Next, I would use the given information to set up equations for Tim and Rick's times. For Tim, we know that he walks half of the distance and runs the other half, so his time can be calculated by adding the time it takes him to walk half the distance (D/2) at speed v_w and the time it takes him to run the other half (D/2) at speed v_r. This can be represented as:

t_Tim = (D/2v_w) + (D/2v_r)

Similarly, for Rick, we know that he walks half the distance and runs the other half, but he walks the distance at a constant speed of v_w and runs at a constant speed of v_r. This can be represented as:

t_Rick = (D/2v_w) + (D/2v_r)

From these equations, we can see that Tim's time is equal to Rick's time, but with an additional factor of (1/v_w - 1/v_r) in front. This means that Tim's time will be less than Rick's time as long as v_r is greater than v_w, which is given in the problem.

To calculate the difference in time, Delta t, between Tim and Rick, we can simply subtract Tim's time from Rick's time:

Delta t = t_Rick - t_Tim

Simplifying this expression by substituting in the equations for t_Tim and t_Rick, we get:

Delta t = (D/2v_w + D/2v_r) - (D/2v_w + D/2v_r) = 0

This means that Tim and Rick will cover the distance in the same amount of time, regardless of the values of v_r and v_w. This can also be verified by plugging in the given special case of v_r = v_w, which would result in both Tim and Rick covering the distance in the same amount of time.

In conclusion, as a scientist, I would approach this problem by carefully defining all variables and units, setting up equations based on the given information