Calculating Separation of Cars in a Convoy Moving at Max Speed

In summary, the problem involves 50 identical cars in a convoy on a long straight level road. The first car starts immediately and each subsequent car starts when its distance from the car ahead becomes 35m. The maximum speed limit is 72km/h and the maximum acceleration and deceleration are 2 m/s^2 and 4 m/s^2 respectively. When all cars are moving at maximum speed, the separation between two adjacent cars is 110m.
  • #1
Ujjwal Basumatary
17
0

Homework Statement


In a convoy on a long straight level road, 50 identical cars are at rest in a queue at equal separation ## 10 m ## from each other. Engine of a car can provide a constant acceleration of ##2 m/s^2##. And brakes can provide a maximum deceleration of ##4m/s^2##. When an order is given to start the convoy, the first car starts immediately and each subsequent car starts when its distance from a car that is immediately ahead becomes ##35m##. Maximum speed limit on this road is ##72km/h##. When an order is given to stop the convoy, the driver of the first car applies brakes immediately and the driver of each subsequent car applies brakes with a certain time delay after noticing the brake light of the front car turn red.

1. When all the cars are moving at the maximum speed, what is the separation between two adjacent cars?

There are other parts to the question but I want to attempt them first before requesting a solution.

Homework Equations


##s=ut + \frac{1}{2}at^2##
##v=u+at##
##v^2=u^2+2as##

where ##s##, ##u##, ##v##, ##t## and ##a## denote displacement, initial velocity, final velocity, time and acceleration respectively.

The Attempt at a Solution


Setting up a coordinate system with the origin at the second car, at ##t=0##, ##S_{1i}=0## and ##S_{2i}=10## denote respectively the initial positions of the second and the first car at the initial moment. According to the problem statement, the displacement of the first car should be ##25m## from its initial position for the second car to start moving.

Using this fact and the equations of motion I get that at ##t=5s##, the first car is at a distance ##35m## from the first. This is the point where the second car starts moving. Taking care of the coordinates and the respective velocity constraints I am getting the separation when both have attained the maximum velocity to be ##85m## whereas the answer given is ##110m##. Where have I gone wrong?
Please help.
Thank you.
 
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  • #2
How did you get 85 m? As this is 25 m shorter than the right answer it looks like you forgot that contribution.

How I would calculate it: 5 seconds less at full speed gives 100 m separation, add the 10 m initial separation: 110 m. No need to keep track of the separation as function of time.
 
  • #3
Ujjwal Basumatary said:
Using this fact and the equations of motion I get that at ##t=5s##, the first car is at a distance ##35m## from the first. This is the point where the second car starts moving. Taking care of the coordinates and the respective velocity constraints I am getting the separation when both have attained the maximum velocity to be ##85m## whereas the answer given is ##110m##. Where have I gone wrong?
Please help.
Thank you.

I just happened to notice that ##72km/h = 20m/s## and that ##110m = 10m + (20m/s \times 5s)##. I wonder if that is a coincidence?
 
  • #4
PeroK said:
I just happened to notice that ##72km/h = 20m/s## and that ##110m = 10m + (20m/s \times 5s)##. I wonder if that is a coincidence?
Got it. Thanks :)
 
  • #5
mfb said:
How did you get 85 m? As this is 25 m shorter than the right answer it looks like you forgot that contribution.

How I would calculate it: 5 seconds less at full speed gives 100 m separation, add the 10 m initial separation: 110 m. No need to keep track of the separation as function of time.
Yeah I missed out that part. Got it anyway. Thanks a lot.
 

Related to Calculating Separation of Cars in a Convoy Moving at Max Speed

1. How is the separation between cars in a convoy calculated?

The separation between cars in a convoy can be calculated by dividing the distance between the cars by the speed at which the convoy is moving. This will give you the time it takes for the convoy to cover the distance between each car.

2. Does the speed of the convoy affect the separation between cars?

Yes, the speed of the convoy does affect the separation between cars. As the speed increases, the distance between cars decreases, resulting in a shorter separation time.

3. What factors can affect the accuracy of the calculated separation between cars?

There are several factors that can affect the accuracy of the calculated separation between cars. These include variations in speed or acceleration, changes in road conditions, and the presence of other vehicles on the road.

4. How can the separation between cars in a convoy be optimized?

The separation between cars in a convoy can be optimized by adjusting the speed and acceleration of the convoy to maintain a consistent and safe distance between cars. Advanced technologies such as sensors and adaptive cruise control can also help optimize the separation between cars.

5. What is the importance of accurately calculating separation between cars in a convoy?

The accurate calculation of separation between cars in a convoy is crucial for ensuring the safety of all vehicles on the road. A proper separation distance allows for enough reaction time in case of sudden stops or changes in road conditions, reducing the risk of accidents and collisions.

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