To solve the problem of determining the distance a car should travel to attain a speed of 180 km/h while accelerating uniformly from rest at a rate of 2.5 m/s², we can follow these steps:
1. Understand the known values:
- Initial velocity ([tex]\( u \)[/tex]) = 0 m/s (since the car starts from rest)
- Acceleration ([tex]\( a \)[/tex]) = 2.5 m/s²
- Final velocity ([tex]\( v \)[/tex]) = 180 km/h
2. Convert the final velocity to consistent units:
- The final velocity is given in kilometers per hour (km/h), which we need to convert to meters per second (m/s).
- To convert from km/h to m/s, use the conversion factor: [tex]\( 1 \text{ km/h} = \frac{1000}{3600} \text{ m/s} \)[/tex].
Therefore,
[tex]\[
v = 180 \times \frac{1000}{3600} \text{ m/s}
\][/tex]
[tex]\[
v = 50 \text{ m/s}
\][/tex]
3. Use the kinematic equation:
- The appropriate kinematic equation to relate the initial velocity, final velocity, acceleration, and distance is:
[tex]\[
v^2 = u^2 + 2as
\][/tex]
- Rearrange the equation to solve for the distance ([tex]\( s \)[/tex]):
[tex]\[
s = \frac{v^2 - u^2}{2a}
\][/tex]
4. Substitute the known values into the equation:
- Since the initial velocity ([tex]\( u \)[/tex]) is 0 m/s:
[tex]\[
s = \frac{v^2}{2a}
\][/tex]
- Substitute [tex]\( v = 50 \text{ m/s} \)[/tex] and [tex]\( a = 2.5 \text{ m/s}^2 \)[/tex]:
[tex]\[
s = \frac{(50)^2}{2 \times 2.5}
\][/tex]
[tex]\[
s = \frac{2500}{5}
\][/tex]
[tex]\[
s = 500 \text{ meters}
\][/tex]
Therefore, the car needs to travel 500 meters to attain a speed of 180 km/h when accelerating uniformly at 2.5 m/s².
