An engineer in a locomotive sees a car stuck
on the track at a railroad crossing in front of
the train. When the engineer first sees the
car, the locomotive is 170 m from the crossing
and its speed is 25 m/s.
If the engineer’s reaction time is 0.68 s,
what should be the magnitude of the minimum deceleration to avoid an accident?
Answer in units of m/s^2
Why did the car get stuck on the track? Did it confuse it for a parking spot? Regardless, let's help the engineer avoid a "train wreck" situation.
To calculate the magnitude of the minimum deceleration needed, we need to consider the distance the train travels during the engineer's reaction time.
The formula to calculate the distance travelled during a constant deceleration is given by:
d = (Initial Velocity * Reaction Time) + (0.5 * Deceleration * Reaction Time^2)
Rearranging the formula, we get:
Deceleration = (2 * (d - (Initial Velocity * Reaction Time))) / Reaction Time^2
Plugging in the given values:
Initial Velocity = 25 m/s
Reaction Time = 0.68 s
Distance = 170 m
Deceleration = (2 * (170 - (25 * 0.68))) / (0.68^2)
After number crunching, we find that the magnitude of the minimum deceleration needed to avoid an accident should be approximately 19.6 m/s^2. Give that car on the track a reason to ride off into the sunset!
To determine the magnitude of the minimum deceleration required to avoid an accident, we can apply the following steps:
Step 1: Convert the given information into SI units (meters and seconds).
Given:
Initial distance (s1) = 170 m
Speed (v0) = 25 m/s
Reaction time (t) = 0.68 s
Step 2: Calculate the distance traveled during the reaction time.
The distance traveled during the reaction time can be calculated using the formula:
distance = speed × time
d1 = v0 × t
d1 = 25 m/s × 0.68 s
d1 = 17 m
Step 3: Calculate the remaining distance to the crossing.
The remaining distance to the crossing can be calculated by subtracting the distance traveled during the reaction time from the initial distance.
d2 = s1 - d1
d2 = 170 m - 17 m
d2 = 153 m
Step 4: Determine the minimum deceleration required.
The minimum deceleration required to avoid an accident can be calculated using the following kinematic equation:
v^2 = u^2 + 2as
where:
v = final velocity (0 m/s as the aim is to stop)
u = initial velocity (25 m/s)
a = deceleration
s = distance (153 m)
0^2 = 25^2 + 2a × 153
0 = 625 + 306a
306a = -625
a = -625 / 306
a ≈ -2.046 m/s^2
However, since acceleration or deceleration cannot be negative in this context, the magnitude of the minimum deceleration is:
|a| ≈ |-2.046| ≈ 2.046 m/s^2
Therefore, the magnitude of the minimum deceleration required to avoid an accident is approximately 2.046 m/s^2.
To find the magnitude of the minimum deceleration required to avoid an accident, we need to calculate the distance the train will travel during the engineer's reaction time and compare it to the distance between the train and the car.
Let's break down the problem:
1. Calculate the distance traveled during the engineer's reaction time:
Distance = speed × time
Distance = 25 m/s × 0.68 s
Distance = 17 m
2. Calculate the remaining distance between the train and the car:
Remaining distance = Distance between the train and the crossing - Distance traveled during reaction time
Remaining distance = 170 m - 17 m
Remaining distance = 153 m
3. Calculate the minimum deceleration required to stop before reaching the car:
The deceleration can be calculated using the following equation:
v^2 = u^2 + 2as
where v is the final velocity (0 m/s), u is the initial velocity (25 m/s), a is the deceleration, and s is the distance to be covered (remaining distance).
Rearranging the equation, we have:
a = (v^2 - u^2) / (2s)
a = (0 - (25 m/s)^2) / (2 × 153 m)
a = -625 m^2/s^2 / 306 m
a = -2.04 m/s^2
The magnitude of the minimum deceleration required to avoid an accident is 2.04 m/s^2 (approximately).