The train, with a speed of 90 km / h, began to move with a deceleration of 0.3 m / s2.

The train, with a speed of 90 km / h, began to move with a deceleration of 0.3 m / s2. Find the speed of the train at a distance of 1 km from the place where it started braking.

Given:
v = 90 km / h = 25 m / s – the initial speed of the train before the start of braking;
a = 0.3 m / s ^ 2 – acceleration with which the train brakes;
S = 1 kilometer = 1000 meters – distance traveled.
It is required to determine v1 (m / s) – the speed of the train through the distance S after the start of braking.
Let us find the time during which the train will pass the distance S:
S = v * t – a * t ^ 2/2;
2 * S = 2 * v * t – a * t ^ 2;
a * t ^ 2 – 2 * v * t + 2 * S = 0;
0.3 * t ^ 2 – 50 * t + 2000 = 0 – we get a quadratic equation.
D = 50 ^ 2 – 4 * 2000 * 0.3 = 2500 – 2400 = 100;
D ^ 0.5 = 100 ^ 0.5 = 10.
t1 = (50 + 10) / (2 * 0.3) = 60 / 0.6 = 100 seconds.
t2 = (50 – 10) / (2 * 0.3) = 40 / 0.6 = 66.6 seconds.
Since both roots of the equation have a positive value, we take the smallest value for the required travel time, that is, t2. Then, the train speed will be:
v1 = v – a * t = 25 – 0.3 * 66.6 = 25 – 19.98 = 5 m / s.
Answer: the train speed will be 5 m / s.