(edit: oops, this should be in college physics)
1. write a model for the speed of the mass in terms of the following quantities: g, m, l, and tension in the lower string.

2. at what speed will the two tensions be equal?

i'm really stuck on this, but i have solved the first couple of parts, such that:

and theta = 30 degrees.

v = √[ gL (3 + 2√3) / 2 ]

Explanation:

Draw a free body diagram of the ball.  There are three forces acting on the ball:

Weight force mg pulling down

Tension force T₁ pulling left

Tension force T₂ pulling at angle θ

d = L/2, so as you found, θ = 30° and r = √3/2 L.

Sum the forces in the y direction:

∑F = ma

T₂ sin 30° − mg = 0

T₂ = 2mg

Sum the forces in the radial direction:

∑F = ma

T₂ cos 30° + T₁ = m v² / r

√3/2 T₂ + T₁ = m v² / (√3/2 L)

3/4 T₂ + √3/2 T₁ = m v² / L

3T₂ + 2√3 T₁ = 4m v² / L

v² = L (3T₂ + 2√3 T₁) / (4m)

v = √[ L (3T₂ + 2√3 T₁) / (4m) ]

Substitute 2mg for T₂:

v = √[ L (6mg + 2√3 T₁) / (4m) ]

If the tensions are equal, then T₁ = 2mg.

v = √[ L (6mg + 4√3 mg) / (4m) ]

v = √[ L (3g + 2√3 g) / 2 ]

v = √[ gL (3 + 2√3) / 2 ]

when jerome paddle in flowing river towards east the net speed of jerome will increase

so the net speed of jerome will be given as

now the speed of paul will be same as it is given as he is running on the ground

so

so here we can say since the speed of jerome is more so he will win the race

now as we know the distance so the time difference is given as

so jerome will reach first and the time difference will be 203.55 s or 3.4 min