You're driving parallel to train tracks at 52.0 km/h when a train passes you, going a steady 91.0 km/h. A track crossing is up ahead, so you begin to speed up at 8.00 m/s2. How far do you drive before you pass the train?

Well, let me calculate that for you while balancing my clown nose on my finger. But first, let's convert all these units to make sure we are not throwing pies at each other.

You're driving at 52.0 km/h, which is approximately 14.4 m/s. The train is moving at 91.0 km/h, or about 25.3 m/s. Now, when you start accelerating at 8.00 m/s², things start to get interesting.

To find out how far you drive before passing the train, we need to figure out the time it takes for your speed to equal or surpass the train's speed. Let's call this glorious moment in time "t".

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 rearrange the equation to solve for time.

25.3 m/s (train's velocity) = 14.4 m/s (your initial velocity) + 8.00 m/s² (your acceleration) × t (time).

So t = (25.3 m/s - 14.4 m/s) / 8.00 m/s², which gives us approximately 1.36 seconds.

Now that we know the time it takes for you to catch up with the train, we can calculate the distance traveled using the formula s = ut + (1/2)at².

s = 14.4 m/s × 1.36 s + (1/2) × 8.00 m/s² × (1.36 s)².

After some silly number crunching, the answer is roughly 13.67 meters.

So, buckle up and enjoy the ride while you pass the train, but make sure not to speed up too fast; otherwise, you might end up joining the circus!

To determine the distance you drive before passing the train, we can follow these steps:

Step 1: Convert the given velocities to meters per second (m/s).
- Your velocity: 52.0 km/h = 52.0 * (1000 m/1 km) * (1 h/3600 s) = 14.44 m/s
- Train velocity: 91.0 km/h = 91.0 * (1000 m/1 km) * (1 h/3600 s) = 25.28 m/s

Step 2: Determine the time it takes for the train to pass you.
- Since the train is traveling at a constant velocity, the time it takes to pass you is the distance traveled divided by the train's velocity.
- Since you are both traveling parallel to the train tracks, the distance traveled by the train is the difference in distances covered by you and the train.
- The time taken by the train to pass you is given by: time = (distance covered by you - distance covered by train) / (train velocity - your velocity)

Step 3: Calculate the distance traveled by you when passing the train given the acceleration.
- Let t be the time taken to pass the train.
- The distance you travel during this time is given by: distance = (initial velocity * t) + (0.5 * acceleration * t^2)

Now let's plug in the values and calculate:

Step 1:
- Your velocity (v1) = 14.44 m/s
- Train velocity (v2) = 25.28 m/s

Step 2:
- Distance traveled by you (d1) = 0 m (since you start at the same point)
- Distance covered by the train (d2) = v2 * t
- Time taken by the train to pass you = t = (d1 - d2) / (v2 - v1)

Step 3:
- Distance traveled by you to pass the train = (v1 * t) + (0.5 * acceleration * t^2)

Now let's calculate the time taken by the train to pass you (t):

t = (d1 - d2) / (v2 - v1)
= (0 - (v2 * t)) / (v2 - v1)
= -(v2 * t) / (v2 - v1)
= -25.28 * t / (25.28 - 14.44)

Let's solve for t:

t = -(25.28 * t) / (10.84)
=> t = 10.84 / 25.28
=> t ≈ 0.429 s

Now, let's calculate the distance you travel to pass the train (distance):

distance = (v1 * t) + (0.5 * acceleration * t^2)
= (14.44 * 0.429) + (0.5 * 8.00 * 0.429^2)

Calculating the expression:

distance = 6.19276 + (0.5 * 8.00 * 0.184041)
= 6.19276 + 0.7363296

Therefore, the distance you drive before passing the train is approximately 6.9291 meters.

To find out how far you drive before passing the train, you can calculate the time it takes for you to catch up with the train. Once you know the time, you can use it to find the distance traveled.

Let's break down the problem:

1. Convert the speeds to m/s:
- Your speed: 52.0 km/h = (52.0 km/h) × (1000 m/1 km) × (1 h/3600 s) = 14.4 m/s
- Train's speed: 91.0 km/h = (91.0 km/h) × (1000 m/1 km) × (1 h/3600 s) = 25.3 m/s

2. Determine the relative speed between you and the train:
Since you're driving parallel to the train, the relative speed is the difference between your speed and the train's speed.
Relative speed = Train's speed - Your speed = 25.3 m/s - 14.4 m/s = 10.9 m/s

3. Calculate the time it takes to catch up with the train:
The equation to calculate time is t = (final velocity - initial velocity) / acceleration.

Given:
Initial velocity (relative speed) = 10.9 m/s
Acceleration = 8.00 m/s^2 (positive because you're speeding up)

t = (0 m/s - 10.9 m/s) / (-8.00 m/s^2)
t = 10.9 m/s / 8.00 m/s^2
t = 1.3625 s (rounded to four decimal places)

4. Calculate the distance traveled:
Distance = (initial velocity × time) + (0.5 × acceleration × time^2)

Given:
Initial velocity (relative speed) = 10.9 m/s
Acceleration = 8.00 m/s^2
Time = 1.3625 s

Distance = (10.9 m/s × 1.3625 s) + (0.5 × 8.00 m/s^2 × (1.3625 s)^2)
Distance ≈ 14.849 m (rounded to three decimal places)

Therefore, you would drive approximately 14.849 meters before passing the train.

8.0 m/s^2 * 1km/1000m * (3600s/hr)^2 = 103680 km/hr^2

after h hours, the distances must be the same:

91.0h = 52.0h + 51840h^2
h = 0.000752315 hours = 2.7 seconds

Or, you may want to convert km/hr to m/s, but the answer is the same.

To get the distance, just plug it into either side of the equation.

That seems pretty fast to catch up, but 8.0 m/s^2 = 28.8 km/hr/s, so after 2.7 seconds, you have accelerated to 52+28.8*2.7=129.76 km/hr, so I guess the time seems reasonable, yeah?