To find the initial velocity of the aircraft, we can use the kinematic equation that relates initial velocity, final velocity, acceleration, and time. The equation is:
[tex]\[ v_f = v_i + a \cdot t \][/tex]
Where:
- [tex]\( v_f \)[/tex] is the final velocity.
- [tex]\( v_i \)[/tex] is the initial velocity.
- [tex]\( a \)[/tex] is the acceleration (in this case, deceleration, so it will be a negative value).
- [tex]\( t \)[/tex] is the time.
We need to find the initial velocity [tex]\( v_i \)[/tex]. We are given:
- Deceleration [tex]\( a = -0.5 \, \text{m/s}^2 \)[/tex] (since the aircraft is slowing down).
- Time [tex]\( t = 8 \, \text{minutes} \)[/tex].
- Final velocity [tex]\( v_f = 160 \, \text{m/s} \)[/tex].
First, we need to convert the time from minutes to seconds:
[tex]\[
t = 8 \, \text{minutes} \times 60 \, \text{seconds/minute} = 480 \, \text{seconds}
\][/tex]
Next, we rearrange the kinematic equation to solve for the initial velocity [tex]\( v_i \)[/tex]:
[tex]\[
v_i = v_f - a \cdot t
\][/tex]
Substitute the given values into the equation:
[tex]\[
v_i = 160 \, \text{m/s} - (-0.5 \, \text{m/s}^2) \cdot 480 \, \text{seconds}
\][/tex]
Since the deceleration is negative, it becomes:
[tex]\[
v_i = 160 \, \text{m/s} + 0.5 \cdot 480
\][/tex]
Calculate the product:
[tex]\[
0.5 \cdot 480 = 240
\][/tex]
Add this to the final velocity:
[tex]\[
v_i = 160 + 240 = 400 \, \text{m/s}
\][/tex]
Therefore, the initial velocity of the aircraft was [tex]\( 400 \, \text{m/s} \)[/tex].
