Final answer:
To calculate the magnitude of the total acceleration of a train at a specific speed, both the tangential and centripetal accelerations must be determined. The total acceleration is found to be approximately 4.47 m/s2 at the speed of 31.7 m/s.
Step-by-step explanation:
The student's question involves computing the magnitude of the total acceleration of a train as it slows down while rounding a bend. To find this, we need to determine both the tangential and centripetal components of acceleration at the moment the train reaches a speed of 31.7 m/s. The total acceleration is the vector sum of these two perpendicular components.
First, we calculate the tangential acceleration by using the initial and final speeds and the time it takes for the train to slow down. The tangential acceleration (at) is constant, and we can find it by using the formula:
at = (final velocity - initial velocity) / time = (22.9 m/s - 54.1 m/s) / 17.6 s = -1.77 m/s2
Next, we find the centripetal acceleration (ac) at the instant when the speed is 31.7 m/s using the formula:
ac = v2 / r = (31.7 m/s)2 / 242.1 m = 4.16 m/s2
Finally, we calculate the total acceleration (atotal) by finding the magnitude of the vector sum of tangential and centripetal accelerations:
atotal = √(at2 + ac2) = √((-1.77 m/s2)2 + (4.16 m/s2)2) = 4.47 m/s2
Therefore, the magnitude of the total acceleration of the train at 31.7 m/s is approximately 4.47 m/s2.