Question regarding motion of vehicles

[kitty89]

1. Question:

You are driving your car (take the direction to be positive) at 20m/s. A van is following you, also at 20m/s. When you see a fog bank ahead of you, you hit the brakes (take this to be the time t=0). The driver of the van sees your brake lights and applies her brakes 1 second later. Both you and the van, when braking, have a constant acceleration of 5m/s^2 magnitude until coming to a stop.

After the van and the car have both stopped, are they further apart, closer together, or the same distance apart than at t=0? How much, in meters, has the distance between them changed?

How far behind the car should the van be at t=0 to avoid a collision?

2. Homework Equations :

Area under the graph of velocity versus time = distance travelled
s=s(initial)+v(initial)*t+(0.5)(a)(t^2)

The Attempt at a Solution

The van and the car are further apart after stopping.
Distance traveled by car after t=0:
(0.5)(4^2)(-5)=40m
Distance traveled by van after t=0:
(4)(20)+(0.5)(-5)(4^2)+20=60m
Change in distance=60m-40m=20m

The car and the van have to be at least 20m apart to avoid collision.

p/s: Is this correct? Thank you very much! =)

 

[anathema]

[/b]

I would like to clarify a few things before providing an answer to your questions. Firstly, it is important to note that in this scenario, we are assuming that the car and van are moving in a straight line and that their velocities are constant before braking. Also, it is not specified whether the van is directly behind the car or if there is some distance between them.

Now, to answer your first question, yes, the car and the van will be further apart after stopping. This is because the car has a head start of 1 second before the van starts braking, so it will travel a greater distance in that time. Using the equations provided, we can calculate the distance traveled by the car and van after braking. The car will travel 40 meters and the van will travel 60 meters, resulting in a change in distance of 20 meters.

For your second question, we need to calculate the initial distance between the car and van at t=0 in order to avoid a collision. This can be done by setting up an equation where the total distance traveled by the car and van is equal to the initial distance between them. Using the equations provided, we get:

40 + 20 = x + 20x = 60/19 = 3.16 meters

Therefore, the van should be at least 3.16 meters behind the car at t=0 to avoid a collision.

I hope this helps clarify the situation and provides a more accurate answer to your questions. Keep in mind that this is just one possible solution and there may be other factors to consider in a real-life scenario. it is always important to consider all variables and make accurate calculations based on the given information.

 

[tomcue]

Yes, your solution is correct. The car and the van are further apart after stopping, and the distance between them has increased by 20m. In order to avoid a collision, the van should be at least 20m behind the car at t=0. This is because both vehicles have the same initial velocity and the same constant acceleration, so the distance between them will continue to increase at a constant rate until they both come to a stop.