To solve the problem of when the automobile overtakes the truck, we can break it down into a few logical steps. We know the truck has a constant acceleration of 4 m/s² and the automobile accelerates at 6 m/s². Both vehicles start from rest, and we need to find out how long it takes for the automobile to catch up to the truck after the truck has covered 150 meters.
Understanding the Motion of Both Vehicles
First, let's establish the equations of motion for both the truck and the automobile. Since both vehicles start from rest, we can use the following kinematic equation:
- Distance (s) = Initial Velocity (u) × Time (t) + 0.5 × Acceleration (a) × Time² (t²)
For the truck:
- Initial Velocity (u) = 0 m/s
- Acceleration (a) = 4 m/s²
- Distance (s) = 150 m
Plugging these values into the equation gives us:
150 = 0 × t + 0.5 × 4 × t²
This simplifies to:
150 = 2t²
From this, we can solve for time (t):
t² = 150 / 2 = 75
t = √75 ≈ 8.66 seconds
Finding the Distance Covered by the Automobile
Now that we know the time it takes for the truck to cover 150 meters, we can find out how far the automobile travels in that same time. Using the same kinematic equation for the automobile:
- Initial Velocity (u) = 0 m/s
- Acceleration (a) = 6 m/s²
We can calculate the distance (s) covered by the automobile in 8.66 seconds:
s = 0 × t + 0.5 × 6 × (8.66)²
This simplifies to:
s = 0.5 × 6 × 75 ≈ 225 meters
Determining the Initial Distance Behind the Truck
Since the automobile travels 225 meters while the truck travels 150 meters, we can find the initial distance the automobile was behind the truck:
Initial Distance = Distance covered by automobile - Distance covered by truck
Initial Distance = 225 m - 150 m = 75 m
Final Summary
In summary, the automobile overtakes the truck after approximately 8.66 seconds, and it was initially 75 meters behind the truck. This problem illustrates the principles of uniformly accelerated motion and how to apply kinematic equations to solve real-world scenarios involving two moving objects.
