A train travels 84 km/h when it reaches a slower train 47 meters ahead traveling the same direction at 24 km/h. If the faster train decelerates at 2.1 m/s^2 while the slower train is at a constant speed, when will they collide? If the train that is faster begins to decelerate at 2.1 m/s^2 while the slower train continues at a constant speed, what willbe the relative speed in which they will collide?
I got 4.9 seconds for the first one and I do not get the last one. Thanks Y'all.
84 km/hr *(1000m/km) /(3600 s/hr)= 23.3 m/s
24 km/hr = 6.67 m/s
d fast train = d
d slow train = d-47
d fast train= d = 23.3 t - .5(2.1) t^2
d slow train = (d-47) = 6.67 t
so
23.3 t - 1.05 t^2 = 6.67 t + 47
1.05 t^2 - 16.6 t + 47 = 0
approximate
2t = 26.6 +/- sqrt (276-188)
2 t = 26.6 +/- 9.4
2t = 36 or 17.2
t = 18 or 8.8
the 8.8 is the first overtaking, the 18 is when the slow one then catches up with the fast one again after it really slows down (assuming they are actually on parallel tracks and do not crash)
To find out when the two trains will collide, we need to calculate the time it takes for the faster train to catch up with the slower train.
First, let's convert the speeds from km/h to m/s.
Speed of the faster train = 84 km/h = (84 * 1000) m/ (60 * 60) s = 23.33 m/s
Speed of the slower train = 24 km/h = (24 * 1000) m/ (60 * 60) s = 6.67 m/s
Now, let's calculate the relative speed between the two trains:
Relative speed = Speed of the faster train - Speed of the slower train
= 23.33 m/s - 6.67 m/s
= 16.66 m/s
To determine when the two trains will collide, we need to find the time it takes for the faster train to cover the initial distance between them (47 meters) plus the additional distance due to the relative speed.
The equation to calculate the time taken is:
Time = Distance / Relative speed
Distance = 47 meters + Initial distance between the two trains
Now, we need to calculate the initial distance between the two trains. Both trains are initially 47 meters apart, but it will take the faster train some time to reach that point.
The equation to calculate the initial distance is:
Initial distance = Initial distance - Distance covered by the slower train in that time
To find the distance covered by the slower train, we use the equation:
Distance = Speed * Time
Now, let's calculate the time the faster train will take to cover the initial distance:
Time = Initial distance / Speed of the faster train
= (47m + Initial distance) / 23.33 m/s
Now, we can rewrite the equation for the initial distance:
Initial distance = (47m + Initial distance) - (Speed of the slower train * Time)
Substituting the value of time:
Initial distance = (47m + Initial distance) - (6.67 m/s * ((47m + Initial distance) / 23.33 m/s))
Simplifying the equation, we get:
Initial distance = 47m + Initial distance - (6.67/23.33) * (47m + Initial distance)
After simplification:
Initial distance = 47m + Initial distance - 0.286 * Initial distance
Combining like terms:
(1 + 0.286) * Initial distance = 47m
Initial distance = 47m / 1.286
Initial distance = 36.58 meters
Now that we have the initial distance, we can calculate the time it takes for the trains to collide:
Time = Distance / Relative speed
= (47m + 36.58m) / 16.66 m/s
= 83.58m / 16.66 m/s
≈ 5.02 seconds
For the second question, we need to find the relative speed at the time of collision when the faster train decelerates at 2.1 m/s^2 while the slower train continues at a constant speed.
Relative speed at collision = Speed of the slower train - Deceleration of the faster train
Relative speed = 6.67 m/s - 2.1 m/s^2
= 4.57 m/s
Therefore, the relative speed at which the two trains will collide is approximately 4.57 m/s.