When a high-speed passenger train traveling at 98 mi/h rounds a bend, the engineer is shocked to see that a locomotive has improperly entered onto the track from a siding 0.42 mi ahead. The locomotive is moving at 16 mi/h. The engineer of the passenger train immediately applies the brakes.

What do you want to know?

To find out how long it takes for the high-speed passenger train to collide with the locomotive, we can use the equation of motion.

First, let's convert the speeds from miles per hour to miles per minute for easier calculations.

The speed of the high-speed passenger train is:
98 miles/hour × (1 hour/60 minutes) = 1.6333 miles/minute (approximately)

The speed of the locomotive is:
16 miles/hour × (1 hour/60 minutes) = 0.2667 miles/minute (approximately)

Now, let's determine the time it takes for the collision to occur. We'll assume that the train starts braking immediately after the engineer sees the locomotive.

Let's define:
d = distance between the train and locomotive when the engineer sees the locomotive (0.42 miles)
Vp = speed of the passenger train in miles per minute (1.6333 miles/minute)
Vl = speed of the locomotive in miles per minute (0.2667 miles/minute)
t = time in minutes

The equation for the distance traveled by an object undergoing uniform deceleration is:
d = (Vo*t) - (0.5*a*t^2)

In this case, the initial velocity of the passenger train (Vo) is Vp, and its acceleration (a) is negative (due to braking). Therefore, the equation becomes:
d = (Vp*t) - (0.5*a*t^2)

For the locomotive, the equation is:
d = (Vl*t)

Since both objects are moving in the same direction, we can set the two equations equal to each other:
(Vp*t) - (0.5*a*t^2) = (Vl*t)

Rearranging the equation, we get:
0.5*a*t^2 - (Vp - Vl)*t + 0 = 0

This is a quadratic equation, where:
a = 0.5*a
b = (Vp - Vl)
c = 0

We can now use the quadratic formula to solve for t:
t = (-b ± sqrt(b^2 - 4*a*c)) / (2*a)

After plugging in the values, we get:
t = (-(Vp - Vl) ± sqrt((Vp - Vl)^2 - 4*(0.5*a)*0)) / (2*0.5*a)

Since both the passenger train and the locomotive are decelerating, we have a < 0. Therefore, we can simplify the equation as:
t = (-(Vp - Vl) ± sqrt((Vp - Vl)^2)) / a

Now we can substitute the known values:
Vp = 1.6333 miles/minute
Vl = 0.2667 miles/minute
d = 0.42 miles

By substituting these values, we can solve for t.