To solve the problem of finding the rocket's velocity 2.5 seconds earlier, we need to use one of the basic kinematic equations that relate acceleration, initial velocity, final velocity, and time. The kinematic equation we utilize is:
[tex]\[ v = u + at \][/tex]
Here,
- [tex]\( v \)[/tex] is the final velocity,
- [tex]\( u \)[/tex] is the initial velocity (or the velocity we need to find, 2.5 seconds earlier),
- [tex]\( a \)[/tex] is the constant acceleration, and
- [tex]\( t \)[/tex] is the time.
We need to rearrange this equation to solve for [tex]\( u \)[/tex], the initial velocity:
[tex]\[ u = v - at \][/tex]
Given data:
- The constant acceleration, [tex]\( a = +14.3 \, m/s^2 \)[/tex],
- The velocity at the later time, [tex]\( v = +55 \, m/s \)[/tex],
- The time difference, [tex]\( t = 2.5 \, s \)[/tex].
Now substitute the values into the rearranged equation:
[tex]\[ u = 55 \, m/s - (14.3 \, m/s^2 \times 2.5 \, s) \][/tex]
First, calculate the product of acceleration and time:
[tex]\[ 14.3 \, m/s^2 \times 2.5 \, s = 35.75 \, m/s \][/tex]
Next, subtract this value from the given later velocity:
[tex]\[ u = 55 \, m/s - 35.75 \, m/s \][/tex]
[tex]\[ u = 19.25 \, m/s \][/tex]
Therefore, the rocket's velocity 2.5 seconds earlier was:
[tex]\[ \boxed{19.3 \, m/s} \][/tex]
Note: The most accurate choice from the given options would be [tex]\( 19.3 \, m/s \)[/tex] (considering possible rounding conventions).
