A car is behind a truck going 18 m/s on the highway. The car's driver looks for an opportunity to pass, guessing that his car can accelerate at 0.6 m/s2 and that he has to cover the 20-m length of the truck, plus 10-m extra space at the rear of the truck and 10 m more at the front of it. In the oncoming lane, he sees a car approaching, probably at the speed limit, 25 m/s (55 mph). He estimates that the car is about 500 m away.

Should he pass the car?

40 = 18t +1/2(.6)t^2

Solve for t.
25 t will be how far the other car travels. Remember the car will be traveling during that time as well.

To determine if the car driver can safely pass the truck without colliding with the oncoming car, we can calculate the time it would take for the car to pass the truck, considering its acceleration and the distance it needs to cover.

1. Calculate the time it would take for the car to accelerate from 0 to its desired speed while passing the truck:
- Initial speed = 0 m/s
- Final speed = 18 m/s (the speed of the truck)
- Acceleration = 0.6 m/s^2

Using the formula: v = u + at (where v is the final speed, u is the initial speed, a is the acceleration, and t is the time), we can rearrange the formula to solve for time:

t = (v - u) / a
t = (18 - 0) / 0.6
t ≈ 30 seconds

2. Calculate the total distance the car needs to cover while passing the truck:
- Length of the truck = 20 m
- Extra space at the rear of the truck = 10 m
- Extra space at the front of the truck = 10 m

Total distance = Length of the truck + Extra space at the rear + Extra space at the front
Total distance = 20 m + 10 m + 10 m
Total distance = 40 m

3. Calculate the distance covered by the car in 30 seconds during acceleration using the following formula:
distance = ut + (1/2)at^2 (where u is the initial speed, t is the time, a is the acceleration, and distance is the distance covered)

distance = 0 * 30 + (1/2) * 0.6 * (30^2)
distance ≈ 270 meters

4. Calculate the remaining distance the car needs to cover after acceleration to pass the truck:
Remaining distance = Total distance - Distance covered during acceleration
Remaining distance = 40 m - 270 m
Remaining distance ≈ -230 meters

Since the remaining distance is negative, it means the car would not have enough distance left to safely pass the truck. The car driver should not attempt to pass in this situation to avoid a potential collision with the oncoming car.

To determine whether the car can safely pass the truck, we need to calculate the time it takes for the car to pass and the distance it will travel during that time. Here's how you can calculate it:

1. Calculate the time it takes for the car to pass the truck:
To pass the truck, the car needs to cover the total length of the truck plus the extra spaces at the rear and front. The total distance can be calculated as follows:
Total Distance = Length of Truck + Extra Space at Rear + Extra Space at Front
Total Distance = 20m + 10m + 10m = 40m

Now, we can calculate the time it takes for the car to cover this distance using the car's acceleration:
Time = (Final Velocity - Initial Velocity) / Acceleration

The initial velocity of the car is the same as the velocity of the truck, which is 18 m/s. The final velocity can be determined by considering the speed at which the car plans to pass the truck. Let's assume the car wants to pass the truck at a speed of 25 m/s (the oncoming car's speed).

Time = (25 m/s - 18 m/s) / 0.6 m/s^2 = 11.67 seconds (approximately)

2. Calculate the distance the car will travel during this time:
To calculate the distance, we multiply the car's initial velocity by the time it takes to pass:

Distance = Initial Velocity * Time
Distance = 18 m/s * 11.67 s = 210 m (approximately)

Now that we have calculated the time and distance, let's check if it is safe for the car to pass the truck.

Considering that the car estimates the oncoming car to be 500 m away, if the car can pass the truck before the oncoming car reaches the car's location, it would be safe to pass.

In this case, the car will travel 210 m during the passing time of 11.67 seconds. Therefore, the minimum distance the car should keep from the oncoming car is:

Minimum Safe Distance = Oncoming Car's Speed * Time
Minimum Safe Distance = 25 m/s * 11.67 s = 291.75 m (approximately)

Since the minimum safe distance is less than the estimated distance of the oncoming car (500 m), it seems safe for the car to pass the truck.