To determine the minimum acceleration that car B needs to avoid an accident while overtaking car A, we first need to analyze the situation step by step. Let's break down the problem using the given speeds and distances.
Understanding the Scenario
We have three cars: A, B, and C. Here are the specifics:
- Car A is moving at a speed of 36 km/h.
- Cars B and C are approaching A from opposite directions at 54 km/h each.
- At a certain moment, the distances from A to B and A to C are both 1 km.
Relative Speeds
First, let's convert the speeds from km/h to m/s for easier calculations:
- Car A: 36 km/h = 10 m/s (since 36 * 1000 / 3600 = 10)
- Car B: 54 km/h = 15 m/s (54 * 1000 / 3600 = 15)
- Car C: 54 km/h = 15 m/s (same as B)
Time to Collision
Since both B and C are 1 km away from A, we need to calculate how long it will take for each car to reach A:
- Time for B to reach A: Distance / Speed = 1000 m / 15 m/s = 66.67 seconds.
- Time for C to reach A: Distance / Speed = 1000 m / 15 m/s = 66.67 seconds.
Overtaking Car A
Car B wants to overtake car A before car C reaches A. To do this, we need to determine how long it will take car B to overtake car A. The overtaking maneuver requires B to accelerate past A.
Distance Covered During Overtaking
Assuming car B needs to cover a distance of at least 100 meters to safely overtake car A, we can set up the equations for the motion of car B:
- Initial speed of B (u) = 15 m/s.
- Final speed of B (v) after overtaking can be assumed to be greater than 15 m/s.
- Distance to overtake (s) = 100 m.
Using the Equation of Motion
We can use the equation of motion:
v² = u² + 2as
Where:
- v = final velocity
- u = initial velocity (15 m/s)
- a = acceleration (what we want to find)
- s = distance (100 m)
Rearranging gives us:
a = (v² - u²) / (2s)
Time Constraint
Car C will reach car A in 66.67 seconds, so car B must complete its overtaking in less time than that. If we assume car B accelerates uniformly, we can also use the time taken to overtake:
Using the equation:
s = ut + (1/2)at²
We can set up the equation for the time taken to overtake:
100 = 15t + (1/2)at²
Finding Minimum Acceleration
To find the minimum acceleration, we can set a reasonable final speed for B, say 25 m/s (which is a safe speed to overtake). Plugging this into our earlier equation:
a = (25² - 15²) / (2 * 100)
a = (625 - 225) / 200 = 400 / 200 = 2 m/s²
Conclusion
Thus, the minimum acceleration that car B needs to avoid an accident while overtaking car A is 2 m/s². This ensures that B can complete the overtaking maneuver before car C reaches car A, thereby avoiding a collision.
