Code (Mini) Golf

Pyth, 27 bytes

.Am<sXsd"_\ /"[1_4Z5)Q._C.z

Demonstration

This code does something very clever and not at all type-safe with X. Check it out below.

Explanation:

.Am<sXsd"_\ /"[1_4Z5)Q._C.z
                               Implicit: Z = 0, Q = eval(input())
                               Q is the initial power.
                         .z    Take all input, as a list of lines.
                        C      Transpose, giving all columns.
                      ._       Form all prefixes.
  m                            Map over the prefixes.
      sd                       Concatenate the prefix.
     X  "_\ /"[1_4Z5)          Change '_' to 1, '\' to -4, ' ' to 0, and '/' to 5.
                               In particular, 'U' is left unchanged.
    s                          Reduce on addition.
                               If all elements were numbers,
                               this results in the total change in power.
                               If there was a 'U', it results in a string.
   <                 Q         If the previous result was a number, this compares
                               it with the initial input to see if the ball is
                               still rolling.
                               If the previous result was a string, this slices off
                               the first Q characters, which always has a truthy
                               result.
.A                             Test whether all of the prefixes mapped to a thruthy
                               result.

Haskell, 111 109 bytes

import Data.List
g"_"=1
g"/"=5
g _= -4 
f n=all(>0).scanl(-)n.map g.fst.span(/="U").(>>=words).transpose.lines

Usage example:

*Main> f 27 "      ____       ____ _   \n   __/    \\     /    U \\  \n__/        \\   /        \\_\n            \\_/           "
True
*Main> f 26 "      ____       ____ _   \n   __/    \\     /    U \\  \n__/        \\   /        \\_\n            \\_/           "
False

How it works:

                            lines  -- split into list of lines at nl
                       transpose   -- transpose
                  (>>=words)       -- turn each line into words (i.e. remove spaces)  
            fst.span(/="U")        -- take all words up to but excluding "U"
         map g                     -- turn each word into the speed modifier
    scanl(-)n                      -- build list of partial sums starting with n
                                   --   note: speed modifiers are negative so we
                                   --   use (-) with scanl to build sums 
all(>0)                            -- return true if all sums are greater than 0                                 

Edit: @user81655 found 2 bytes to save. Thanks!

Ruby, 104 87 characters

->s,t{t.lines.map(&:bytes).transpose.map{|o|(c=o.max)==85||s<0?break: s+=c*3%14-6}
s>0}

Sample run:

2.1.5 :001 > track = '      ____       ____ _   
2.1.5 :002'>    __/    \     /    U \  
2.1.5 :003'> __/        \   /        \_
2.1.5 :004'>             \_/           
2.1.5 :005'> '
 => "      ____       ____ _   \n   __/    \\     /    U \\  \n__/        \\   /        \\_\n            \\_/           \n" 

2.1.5 :006 > ->s,t{t.lines.map(&:bytes).transpose.map{|o|(c=o.max)==85||s<0?break: s+=c*3%14-6};s>0}[27, track]
 => true 

2.1.5 :007 > ->s,t{t.lines.map(&:bytes).transpose.map{|o|(c=o.max)==85||s<0?break: s+=c*3%14-6};s>0}[26, track]
 => false