You are driving on a very long, very straight 2 lane road, stuck behind a lorry which travels at a constant 75 km/hour. You want to overtake the lorry, which has a length of 12.0 m. You car is 3.8 m long. To overtake the lorry, and to do this safely, you need to move to the other lane when the front of your car is no less than 10 m behind the lorry. You should return to the left hand lane, when the rear of your car is 10 m ahead of the lorry. Your car has an acceleration of 2.0 ms-2.

(a) What is the minimum distance you need to overtake the lorry ?
(b) How fast will you be going when you move back to the left hand lane?
(c) Suppose that the cars coming in the opposite direction are travelling no faster than 100 km/hour. Determine the minimum distance ahead that the road must be clear of oncoming cars before you start to pass.

To solve this problem, we need to consider the distances, speeds, and acceleration involved in overtaking the lorry safely. Let's break down each part of the problem.

(a) What is the minimum distance you need to overtake the lorry?

To overtake the lorry safely, you need to move to the other lane when the front of your car is no less than 10 m behind the lorry. Therefore, the distance you need to cover to overtake the lorry is the sum of the lengths of both vehicles plus the safety distance on either end.

The distance needed to overtake the lorry is:
Distance = Length of Lorry + Length of Car + Safety Distance
Distance = 12.0 m + 3.8 m + 10 m + 10 m
Distance = 35.8 m

So, the minimum distance you need to overtake the lorry is 35.8 meters.

(b) How fast will you be going when you move back to the left-hand lane?

To determine the speed at which you will be going when moving back to the left-hand lane, we need to calculate the time it takes to overtake the lorry. To calculate the time, we divide the change in velocity (speed) by the acceleration of your car.

Change in Velocity = Final Velocity - Initial Velocity
Final Velocity = Speed of Lorry = 75 km/hour = (75 * 1000) / 3600 = 20.83 m/s (converted from km/h to m/s)
Initial Velocity = 0 m/s (assuming your car starts from rest)

Using the formula: Time = Change in Velocity / Acceleration
Time = 20.83 m/s / 2.0 m/s^2
Time = 10.415 s

Now, let's calculate the speed at which you will be going when moving back to the left-hand lane using the formula: Final Velocity = Initial Velocity + (Acceleration * Time)
Final Velocity = 0 m/s + (2.0 m/s^2 * 10.415 s)
Final Velocity = 20.83 m/s

So, you will be traveling at a speed of 20.83 m/s when moving back to the left-hand lane.

(c) Determine the minimum distance ahead that the road must be clear of oncoming cars before you start to pass.

To make sure it is safe to pass the lorry, we need to consider the distance required to complete the overtaking maneuver before oncoming cars come within a dangerous proximity. We'll assume the oncoming cars are traveling at a maximum speed of 100 km/hour.

First, we need to convert the speed of oncoming cars to meters per second:
Oncoming Cars Speed = 100 km/hour = (100 * 1000) / 3600 = 27.78 m/s

Now, let's calculate the time it takes for oncoming cars to reach the position where you start the overtaking maneuver.

Time = Distance / Oncoming Cars Speed = Distance / 27.78 m/s

We already calculated the distance needed to overtake the lorry (35.8 m). Therefore:
Time = 35.8 m / 27.78 m/s
Time = 1.29 s

To ensure there is a safe distance ahead, we need to consider the time it takes to complete the overtaking maneuver. We calculated the total time to overtake the lorry in part (b) as 10.415 s.

Therefore, the minimum distance ahead that the road must be clear of oncoming cars is the sum of the distance traveled during overtaking and the distance traveled by oncoming cars during the time it takes to overtake.

Distance ahead = Oncoming Cars Speed * (Time to overtake + Time for oncoming cars to reach overtaking position)
Distance ahead = 27.78 m/s * (10.415 s + 1.29 s)
Distance ahead = 27.78 m/s * 11.705 s
Distance ahead = 325.0169 m

So, the road must be clear of oncoming cars for a minimum distance of 325.0169 meters ahead before you start to pass the lorry.

To solve this problem, we need to determine the minimum distance required to overtake the lorry, the speed at which you'll be going when moving back to the left-hand lane, and the minimum distance ahead that the road must be clear of oncoming cars before starting to pass.

(a) To find the minimum distance needed to overtake the lorry:
- The lorry's length is 12.0 m.
- The distance from the front of your car to the lorry when you can start overtaking is 10 m.
- To calculate how long it takes for your car to reach the front of the lorry, we will use the formula:
Time = (final velocity - initial velocity) / acceleration

Since your car is initially traveling at a constant speed of 0 km/hr, the initial velocity is 0 km/hr.
The final velocity will be the speed at which you overtake the lorry, denoted v_final.
The acceleration is given as 2.0 m/s^2.

Rearranging the formula, we have:
Time = (v_final - 0) / 2.0 m/s^2

- The distance traveled during this time can be calculated using the formula:
Distance = (initial velocity * Time) + (0.5 * acceleration * Time^2)

Rearranging the formula, we have:
Distance = (0 * Time) + (0.5 * 2.0 * Time^2)

- The total distance needed to overtake the lorry is the sum of the distance traveled during the time it takes to reach the front of the lorry and the length of the lorry itself:
Total Distance = Distance + lorry length

Rearranging the formula, we have:
Total Distance = 0.5 * 2.0 * Time^2 + 12.0

Now, let's calculate the minimum distance needed to overtake the lorry.

(b) To find the speed at which you'll be going when moving back to the left-hand lane:
- We know that your car's acceleration is 2.0 m/s^2.
- The time it takes for your car to overtake the lorry can be calculated using the formula:
Time = (v_final - initial velocity) / acceleration

Rearranging the formula, we have:
Time = (v_final - 0) / 2.0 m/s^2

- The speed at which you'll be going when moving back to the left-hand lane is the final velocity, denoted v_final.

Now, let's calculate the speed at which you'll be going when moving back to the left-hand lane.

(c) To determine the minimum distance ahead that the road must be clear of oncoming cars before starting to pass:
- The maximum speed of the oncoming cars is given as 100 km/hr.
- To convert this speed into m/s, we can divide by 3.6 (since 1 km/hr = 1000 m / (60 * 60) s).
- The distance the oncoming cars will travel during the time it takes for your car to overtake the lorry is the sum of the length of the lorry and the minimum distance needed to overtake.
- Subtracting this distance from the distance ahead that the road must be clear of oncoming cars will give us the minimum distance required.

Now, let's calculate the minimum distance ahead that the road must be clear of oncoming cars before starting to pass.

Please provide the speed at which you want to overtake the lorry (v_final) for me to continue with the calculations.