Two trains are heading toward each other on the same track with velocities v1 and v2 respectively. The drivers,noticing this when the trains are at distance x apart, apply brakes, giving same acceleration y to each train. The necessary condition for collision is?
The necessary condition for a collision between the trains is that they must come to a stop before they reach each other. In other words, they must both reach a speed of 0 m/s before they cover the remaining distance between them.
To determine the necessary condition for the collision, we can start by analyzing the motion of each train using the equations of motion.
First, let's consider Train 1. We know that its initial velocity is v1 and its acceleration is y. We need to find the time it takes for Train 1 to come to a stop.
Using the equation of motion v = u + at, where:
- v is the final velocity (0 m/s in this case),
- u is the initial velocity (v1),
- a is the acceleration (y), and
- t is the time.
We can rearrange the equation to solve for time:
t = (v - u) / a
Similarly, for Train 2, the initial velocity is v2, and the acceleration is also y. We can use the same equation to find the time it takes for Train 2 to come to a stop.
Now, let's consider the distances covered by each train during this time. Train 1 will cover a distance equal to the initial distance between the trains, x. Train 2 will cover a distance that depends on its initial velocity and the time it takes to come to a stop.
Using the equation of motion s = ut + (1/2)at^2, where:
- s is the distance covered,
- u is the initial velocity,
- t is the time, and
- a is the acceleration (y).
For Train 1, the distance covered, s1, is given by:
s1 = x
For Train 2, the distance covered, s2, is given by:
s2 = v2 * t + (1/2) * y * t^2
Now, for the trains to collide after they stop, the total distance covered by Train 1 and Train 2 must be equal to the initial distance between them, x.
By equating the distances covered by the two trains:
x = s1 + s2
x = x + v2 * t + (1/2) * y * t^2
We can then solve this equation to determine the necessary condition for collision. Equating the coefficients of the terms on both sides:
1 = v2 / y * t + (1/2) * t^2
Simplifying the equation:
t^2 + 2 * (v2 / y) * t - 2 = 0
This equation is a quadratic equation in terms of time, t. For a collision to occur, this equation must have real and positive solutions for time. Mathematically, this condition for real and positive solutions is given by the discriminant of the quadratic equation, which is:
D = (2 * (v2 / y))^2 - 4 * 1 * (-2)
D = (4 * (v2 / y)^2 + 8)
For a collision to occur, the discriminant D must be greater than or equal to 0.
Therefore,
4 * (v2 / y)^2 + 8 ≥ 0
Simplifying the inequality:
(v2 / y)^2 ≥ -2
Since the square of a real number is always greater than or equal to 0, the inequality holds true for all values of (v2 / y). Therefore, the necessary condition for collision is that v2/y can be any real number.
In simpler terms, for the trains to collide, the ratio of Train 2's initial velocity (v2) to the acceleration applied to both trains (y) can be any value, implying that Train 2 can have any initial velocity compared to Train 1's velocity and still result in a collision if both trains apply the same acceleration and come to a stop.
[v1*t -(a/2)t^2] + [v2*t - (a/2)t^2] = x
(v1 +v2)*t - a t^2 = x
a is the acceleration rate.
The time t to collision depends upon the deceleration rate a as well as v1+v2 and x.
If there is no real solution to the quadratic, they do not collide.