Sure, let's solve the problem.
Question:
A train travels at a speed of 60 km/h. Immediately after passing a red signal, it starts to come to a complete stop within a distance of 150 meters. Determine its acceleration.
Solution:
1. Initial Information:
- Initial velocity ([tex]\(u\)[/tex]) = 60 km/h
- Stopping distance ([tex]\(s\)[/tex]) = 150 meters
2. Convert the initial velocity from km/h to m/s:
- To convert km/h to m/s, you can use the formula:
[tex]\[
\text{Velocity in m/s} = \left( \frac{\text{Velocity in km/h} \times 1000}{3600} \right)
\][/tex]
- Plugging in the values:
[tex]\[
u = \frac{60 \times 1000}{3600} = 16.67 \text{ m/s}
\][/tex]
3. Use the kinematic equation to find the acceleration:
- The kinematic equation which relates initial velocity, final velocity, acceleration, and distance is:
[tex]\[
v^2 = u^2 + 2as
\][/tex]
- Here, final velocity ([tex]\(v\)[/tex]) is 0 because the train comes to a stop.
- Rearrange to solve for acceleration ([tex]\(a\)[/tex]):
[tex]\[
0 = u^2 + 2as \implies a = -\frac{u^2}{2s}
\][/tex]
- Substitute the known values into the equation:
[tex]\[
a = -\frac{(16.67)^2}{2 \times 150} \approx -0.93 \text{ m/s}^2
\][/tex]
Answer:
The acceleration of the train is approximately [tex]\(-0.93 \text{ m/s}^2\)[/tex].
This solution accounts for the initial velocity conversion and the application of the kinematic equation to determine the train's deceleration as it stops after passing the red signal.
