To find the initial velocity of the train when it starts to decelerate to a stop, we can use the kinematic equation:
\[ v^2 = u^2 + 2a s \]
where:
- \( v \) is the final velocity (0 m/s, since the train comes to rest),
- \( u \) is the initial velocity (the value we want to find),
- \( a \) is the acceleration (which will be negative since it's deceleration, so \( a = -1.0 , \text{m/s}^2 \)),
- \( s \) is the distance over which the deceleration occurs (513 m).
Substituting the known values into the equation:
\[ 0 = u^2 + 2(-1.0)(513) \]
This simplifies to:
\[ 0 = u^2 - 1026 \]
Rearranging gives:
\[ u^2 = 1026 \]
Taking the square root of both sides:
\[ u = \sqrt{1026} \]
Calculating that gives:
\[ u \approx 32.01 , \text{m/s} \]
Therefore, the initial velocity of the train when the acceleration first began was approximately 32.01 m/s.