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 can use the equations of motion to determine the distances required to overtake the lorry and the final speed when returning to the left lane.
First, let's calculate the time it takes for the front of your car to reach the position 10 m behind the lorry. We can use the acceleration formula:
v = u + at
Where:
v = final velocity
u = initial velocity
a = acceleration
t = time
Since the initial velocity is 0 m/s (as we start from rest), and the acceleration is given as 2.0 m/s^2, we can rearrange the formula to find time:
t = (v - u) / a
Substituting the values, we get:
t = (0 - 75) / -2.0
t = 37.5 s
This means it takes 37.5 seconds for the front of your car to reach the position 10 m behind the lorry.
(a) To find the minimum distance needed to overtake the lorry, we need to calculate the distance covered during this time. We can use the equation of motion to calculate distance:
s = ut + (1/2)at^2
Where:
s = distance
u = initial velocity
t = time
a = acceleration
Since the initial velocity is 0 m/s, the equation simplifies to:
s = (1/2)at^2
Substituting the values, we get:
s = (1/2)(-2.0)(37.5)^2
s = 1406.25 m
Therefore, the minimum distance needed to overtake the lorry is 1406.25 meters.
(b) To calculate the final speed when moving back to the left lane, we can use the equation of motion:
v^2 = u^2 + 2as
Where:
v = final velocity
u = initial velocity
a = acceleration
s = distance
In this case, the initial velocity is 0 m/s, the acceleration is 2.0 m/s^2, and the distance is 10 m (length of your car). Rearranging the equation, we have:
v^2 = 2as
v^2 = 2 * 2.0 * 10
v^2 = 40
Taking the square root of both sides, we get:
v ≈ 6.32 m/s
Therefore, when moving back to the left lane, your speed will be approximately 6.32 meters per second.
(c) To determine the minimum distance ahead that the road must be clear of oncoming cars before you start to pass, we need to consider the time it takes for the rear of your car to pass the lorry. Using the same equation as in part (a), we can calculate this time:
t = (v - u) / a
Where:
v = final velocity
u = initial velocity
a = acceleration
t = time
Since the initial velocity is 0 m/s, the final velocity is 6.32 m/s, and the acceleration is 2.0 m/s^2, we have:
t = (6.32 - 0) / 2.0
t = 3.16 s
Now, considering the speed of the oncoming cars, which is given as 100 km/h, we can convert it to m/s by dividing by 3.6 (1 km/h = 1/3.6 m/s):
Speed of oncoming cars = 100 / 3.6
Speed of oncoming cars ≈ 27.78 m/s
The minimum distance ahead that the road must be clear of oncoming cars is determined by the distance traveled by the rear of your car during the time it takes to pass the lorry. We can use the equation of motion to calculate this distance:
s = ut + (1/2)at^2
Using the same values for initial velocity, time, and acceleration as in part (c), we have:
s = 0 * 3.16 + (1/2) * 2.0 * (3.16)^2
s ≈ 10 m
Therefore, the road must be clear of oncoming cars at least 10 meters ahead before you start to pass.