Final answer:
To avoid an accident, The engineer must decelerate the train to a stop within the remaining distance of 291.9m. The magnitude of the minimum deceleration needed to avoid an accident is 18 m/s².
Step-by-step explanation:
To avoid an accident, the engineer needs to bring the train to a stop before reaching the car stuck on the track. The engineer's reaction time is 0.45 s, during which the train continues to move at its initial velocity. The distance traveled during the reaction time is given by the equation: distance = velocity × time. Substituting the given values, the distance traveled during the reaction time is (18 m/s)(0.45 s) = 8.1 m.
The engineer must decelerate the train to a stop within the remaining distance of 300 m - 8.1 m = 291.9 m. The deceleration is given by the equation: deceleration = change in velocity / time. The change in velocity is the difference between the initial velocity (18 m/s) and the final velocity (0 m/s). The time is the reaction time plus the time it takes to decelerate to a stop.
Let's calculate the time it takes to decelerate to a stop first.
Using the equation for the final velocity during deceleration: final velocity = initial velocity + acceleration × time, we can solve for the time required to come to a stop: 0 m/s = 18 m/s + acceleration × time.
Rearranging the equation gives: time = -18 m/s / acceleration.
Substituting the given values, we have: time = -18 m/s / acceleration.
Now, we can substitute the time into the equation for the distance: distance = (18 m/s)(0.45 s) + (1/2)(acceleration)(time^2).
Substitute the known values and solve for the acceleration: 291.9 m = 8.1 m + (1/2)(acceleration)(-18 m/s / acceleration)^2.
Simplifying the equation gives: 283.8 m = (1/2)(-18 m/s)(-18 m/s / acceleration).
Solving for acceleration, we get: acceleration = -18 m/s^2.
Therefore, the magnitude of the minimum deceleration needed to avoid an accident is 18 m/s².