Let s = "My car does not start."
Let w = "I will make it to work on time."

I) If my car does not start, then I will not make it to work on time.

That's ~s->~w

s  w  ~s  ~w  ~s -> ~w
T  T   F   F      T
T  F   F   T      T
F  T   T   F      F
F  F   T   T      T

II) If my car does not start, then I will make it to work on time.

That's ~s -> w
 
s  w  ~s  ~s -> w
T  T   F     T
T  F   F     T
F  T   T     T
F  F   T     F


III) Either my car starts or I will make it to work on time.

That's s V w

s  w  s V w
T  T    T
T  F    T    
F  T    T    
F  F    F    

II and III are equivalent because they have the same truth table,
TTTF.  

Or you can prove it because 

(p->q) <=> (~pVq) 

Let p = ~s and q = w, then

(~s->w) <=> (~(~s) V w)

(~s->w) <=> (s V w)

Edwin