I need help from with a problem from Yale Open Course
Romeo is at x=0m at t=0s when he sees juliet at x=6m.
a) romeo begins to run towards her at v = 5 m/s. Juliet, in turn begins to accelerate towards him at a =-2m/s^2. when and where will they cross?
b) suppose, instead, that juliet moved away from romeo with positive acceleration a. Find a(max), the maximum acceleration for which romeo can catch up with her. for this case find the time t of their meeting. show that for smaller values of a these star-crossed lovers cross twice.

Question

I need help from with a problem from Yale Open Course
Romeo is at x=0m at t=0s when he sees juliet at x=6m.
a) romeo begins to run towards her at v = 5 m/s. Juliet, in turn begins to accelerate towards him at a =-2m/s^2. when and where will they cross? 
b) suppose, instead, that juliet moved away from romeo with positive acceleration a. Find a(max), the maximum acceleration for which romeo can catch up with her. for this case find the time t of their meeting. show that for smaller values of a these star-crossed lovers cross twice.

amistre64 · Answer

$$a=\frac{2\Delta x}{t^2}$$

amistre64 · Answer

its my first thought :)

amistre64 · Answer

5+2 = 7m/s; to cover 6 meters

amistre64 · Answer

$$t=\sqrt{\frac{2\Delta x}{7}}$$maybe?

amistre64 · Answer

for the other one: $$R=\Delta x-6, ~J=\Delta x$$oy, this is much easier to think about in mathing notations ...

amistre64 · Answer

$$R=\frac12(5)t^2+v_ot=\Delta x-6$$
$$J=\frac12(2)t^2+v_ot=\Delta x$$equate the delta xs to find the time

fwizbang · Answer

Lets set this up more carefully.....
Romeo's velocity is constant and he starts at x=0, so his position at time t is

$$x = (5 m/s) t$$

Juliet starts at rest at x=6 m and accelerates at -2 m/s^2, so her position at time t is

$$x= 6 m  + 0.5*a t^2$$

with a=-2 m/s^2.

When they meet, they're at the same place, so set the two  expressions for x equal to one another and solve for t with the quadratic formula. (Presumably you want the solution with positive t.)Once you have t, put it back into either equation for x to find where they meet.

To find the answer for the second question, solve the quadratic without putting a number in for a. For what values of a will the discriminant be negative?  Finally, 
when the discriminant is positive, there will be two solutions for t due to the plus/minus in the quadratic equation. Show that both values are positive.

The interpretation of the smaller solution is that Juliet is originally moving slowly so that when romeo first catches up to her, he is moving faster than she is and passes her by.
Juliette continues to speed up until she is traveling faster than Romeo, and the larger solution is when she catches back up to him.

Ain't love grand?