A bullet train is traveling at 23.3 m/s when it approaches a slower train 47 meters ahead traveling in the same direction at 6.67 m/s.
If the faster train begins decelerating at 2.1m/s^2 while the slower train continues at constant speed, how soon will they collide?
I got 3.68 seconds.Using the quadriatic formula.
If the faster train begins decelerating at 2.1m/s^2 while the slower train continues at constant speed, at what relative speed will they collide?
I got 8.5 m/s from subtracting 15.2 (what I got from the quadriatic formula)from 6.67 (slower train).
To find out how soon the two trains will collide, we need to calculate the time it takes for the faster train to travel the distance between them. Let's break down the problem step by step:
1. Determine the relative speed between the two trains:
- The velocity of the faster train is 23.3 m/s.
- The velocity of the slower train is 6.67 m/s.
- The relative speed is obtained by subtracting the velocity of the slower train from the velocity of the faster train:
Relative speed = 23.3 m/s - 6.67 m/s = 16.63 m/s.
2. Determine the distance between the two trains:
- The slower train is 47 meters ahead of the faster train.
3. Use the equation of motion to determine the time it takes for the faster train to travel the distance between them:
- The equation to calculate distance is: Distance = Initial velocity * Time + 0.5 * Acceleration * Time^2.
- In this case, the initial velocity is 23.3 m/s, the acceleration is -2.1 m/s^2 (negative because it's decelerating), and the distance is 47 meters.
- Let's rearrange the equation to solve for time: -2.1 * t^2 + 23.3 * t - 47 = 0.
- We can solve this quadratic equation using the quadratic formula.
Using the quadratic formula:
t = (-b ± √(b^2 - 4ac)) / (2a)
In our case:
a = -2.1,
b = 23.3,
c = -47.
Plugging these values into the quadratic formula:
t = (-23.3 ± √(23.3^2 - 4(-2.1)(-47))) / (2 * -2.1)
Simplifying further:
t = (-23.3 ± √(542.89 + 418.8)) / (-4.2)
t = (-23.3 ± √(961.69)) / (-4.2)
t = (-23.3 ± 31.02) / (-4.2)
Now, we have two possible values for time:
t1 = (-23.3 + 31.02) / (-4.2) = 1.71 seconds
t2 = (-23.3 - 31.02) / (-4.2) ≈ -3.68 seconds
Since we are dealing with time, the negative solution doesn't make sense in this context. Therefore, the time it will take for the two trains to collide is approximately 1.71 seconds.
To find the relative speed at the time of the collision, we need to subtract the deceleration from the initial relative speed.
Relative speed at the time of collision = Initial relative speed - Deceleration
Relative speed at the time of collision = 16.63 m/s - 2.1 m/s^2 ≈ 14.53 m/s.
Therefore, the relative speed at the time of collision will be approximately 14.53 m/s.