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 will be the relative speed in which they will collide?
I got 8.5 seconds for the first one and 3.68 seconds for the last one. Thanks Y'all.
To solve the first question, we need to find the time it takes for the faster train to catch up to the slower train.
Step 1: Convert the speeds to meters per second (m/s).
- The speed of the faster train is 84 km/h, which is equal to 84,000 meters per hour or 23.33 meters per second (m/s).
- The speed of the slower train is 24 km/h, which is equal to 24,000 meters per hour or 6.67 meters per second (m/s).
Step 2: Find the relative velocity between the two trains.
- The relative velocity can be calculated by subtracting the speed of the slower train from the speed of the faster train.
- Relative velocity = 23.33 m/s - 6.67 m/s = 16.66 m/s.
Step 3: Find the time it takes for the faster train to cover the distance between them.
- The distance between the trains is 47 meters.
- Time = Distance / Relative velocity = 47 m / 16.66 m/s ≈ 2.82 seconds.
Therefore, the faster train will collide with the slower train after approximately 2.82 seconds.
To solve the second question, we need to find the relative speed at the time of collision when the faster train decelerates.
Step 1: Calculate the deceleration of the faster train.
- The faster train decelerates at a rate of 2.1 m/s^2.
Step 2: Find the time it takes for the faster train to come to a stop.
- Using the formula v = u + at, where v is the final velocity, u is the initial velocity, a is the acceleration, and t is the time, we can calculate the time it takes for the train to stop.
- Final velocity (v) = 0 m/s (since the train comes to a stop)
- Initial velocity (u) = 23.33 m/s
- Acceleration (a) = -2.1 m/s^2 (negative sign because of deceleration)
- 0 = 23.33 m/s + (-2.1 m/s^2) * t
- Solving the equation, we find t ≈ 11.1 seconds.
Step 3: Find the relative speed at the time of collision.
- The relative speed is given by the relative velocity at the time of collision when the faster train stops decelerating and the slower train continues at a constant speed.
- The relative velocity is 16.66 m/s (as calculated before).
- Since the faster train comes to a stop, the relative speed at the time of collision is equal to the speed of the slower train, which is 6.67 m/s.
Therefore, the relative speed at the time of collision will be approximately 6.67 m/s.