You are traveling at a constant speed υM, and there is a car in front of you traveling with a speed υA You notice that υM>υA, so you start slowing down with a constant acceleration a when the distance between you and the other car is x.
What relationship between a and x determines whether or not you run into the car in front of you?
Express your answer in terms of the variables υA, υM, and a.
Well, if we want to determine whether or not you run into the car in front of you, we need to consider the time it takes for you to come to a stop.
Let's set up a little equation to help us out:
x = υM*t + 0.5*a*t^2
In this equation, x represents the distance between you and the other car, t represents the time it takes for you to come to a stop, υM represents your initial speed, and a represents your constant deceleration.
To avoid crashing into the car, you need to make sure that t is greater than the time it takes for the other car to move a distance x. This means that the car in front of you needs to travel a distance x in less time than it takes for you to stop.
The time it takes for the other car to move a distance x is given by:
tA = x / υA
So, in order to avoid running into the car in front of you, you need to make sure that t is greater than tA:
t > tA
Now, let's substitute the equation for t into our inequality:
υM/a > x/υA
Rearranging a bit, we find:
a < υA*υM/x
So, the relationship between a and x that determines whether or not you run into the car in front of you is:
a < υA*υM/x
And remember, always drive carefully and keep a safe distance from other vehicles on the road! Stay clown-safe! 🤡
To determine if you will run into the car in front of you, you need to consider the relative speeds and the distance between the two vehicles.
First, let's define the variables:
υM - Your constant speed
υA - Speed of the car in front of you
a - Your constant deceleration (negative acceleration)
x - Distance between you and the car in front of you
If you want to avoid running into the car in front of you, your deceleration needs to be sufficient to bring your speed to be less than or equal to the speed of the car in front of you before you reach it.
To find the relationship between a and x, we can use the kinematic equation:
υ^2 = υ0^2 + 2aΔx
Where:
υ is your final speed (which should be equal to or less than υA)
υ0 is your initial speed (υM)
a is your constant deceleration (negative acceleration)
Δx is the change in distance between you and the car in front of you (x)
Rearranging the equation, we get:
2aΔx = υ^2 - υ0^2
Since you want your final speed to be equal to or less than υA, we can rewrite this as:
2aΔx = υA^2 - υM^2
Now, the condition for you to avoid running into the car in front of you is that the left side of the equation is negative or zero, meaning your deceleration is sufficient to decrease your speed to υA or below before reaching the car. Therefore, the relationship between a and x is:
aΔx ≤ (υA^2 - υM^2) / 2
If this condition is satisfied (aΔx is less than or equal to the right side of the equation), then you will not run into the car in front of you. If the condition is not satisfied (aΔx is greater than the right side of the equation), you will eventually collide with the car.