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).
The relative speed is 16.63m/s, and the relative deacceleration is 2.1m/s^2
47=16.63t+1/2 2.1 t^2
putting it in standard form..
1.1t^2+16.63t-47=0
t= (-16.63+-sqrt(277+207)/2.2
I don't get your time for the first.
Relative speed can be found knowing the correct time by..
v=initialrelatlivespeed*time-1/2 2.1 t^2
To find out when the two trains will collide, we can use the equation of motion. Let's denote:
- The initial velocity of the faster train as v1 = 23.3 m/s,
- The initial velocity of the slower train as v2 = 6.67 m/s,
- The acceleration of the faster train as a1 = -2.1 m/s^2 (negative because it's decelerating),
- The distance between the two trains as d = 47 m,
- The relative velocity between the two trains as v_rel.
First, let's determine when the faster train comes to a stop (time t1):
Using the equation of motion,
v1 = 0 m/s (final velocity when the faster train stops),
a1 = -2.1 m/s^2 (deceleration of the faster train),
t1 = (v1 - 0)/a1,
t1 = -v1/a1,
t1 = -23.3 m/s / (-2.1 m/s^2),
t1 ≈ 11.1 seconds.
Now, let's calculate the relative distance traveled by the slower train during t1 (d_rel):
d_rel = v2 * t1,
d_rel = 6.67 m/s * 11.1 seconds,
d_rel ≈ 74.17 meters.
Since the distance traveled by the slower train (74.17 meters) is greater than the initial distance between the two trains (47 meters), we can conclude that the collision occurs before the faster train comes to a stop.
To find out when the two trains will collide, we need to find the time when the relative distance becomes equal to the initial distance (d_rel = d):
Let t2 represent the time when the relative distance is equal to the initial distance (47 m).
Using the equation of motion for the slower train:
d_rel = v2 * t2,
47 m = 6.67 m/s * t2,
t2 = 47 m / 6.67 m/s,
t2 ≈ 7.04 seconds.
Therefore, the two trains will collide approximately 7.04 seconds after the faster train applied deceleration.
To calculate the relative speed at the time of collision, we need to find the final velocity of the faster train (v_f1) just before the collision:
Using the equation of motion:
v_f1 = v1 + a1 * t2,
v_f1 = 23.3 m/s + (-2.1 m/s^2) * 7.04 s,
v_f1 ≈ 8.474 m/s.
The relative speed at the time of collision is the difference between the final velocity of the faster train and the initial velocity of the slower train:
v_rel = v_f1 - v2,
v_rel ≈ 8.474 m/s - 6.67 m/s,
v_rel ≈ 1.804 m/s.
Therefore, the two trains will collide at a relative speed of approximately 1.804 m/s.