Question

A train is brought to rest in a distance of 513 m using a constant acceleration of 1.0 m/s2. What was the velocity of the train when the acceleration first began?

Answer 1

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**.