To find the distance the train will need to travel in order to reach a velocity of 21.0 m/s with a uniform acceleration of 2.30 m/s², we can use the following kinematic equation:
v² = u² + 2as
where: v = final velocity (21.0 m/s) u = initial velocity (0 m/s, since the train starts at rest) a = acceleration (2.30 m/s²) s = distance
Rearranging the equation, we have:
s = (v² - u²) / (2a)
Substituting the given values:
s = (21.0² - 0²) / (2 × 2.30) s = (441.0) / (4.60) s ≈ 95.87 meters
Therefore, the train will need to travel approximately 95.87 meters to obtain a velocity of 21.0 m/s.