You're driving parallel to train tracks at 40.0 km/h when a train passes you, going a steady 105 km/h. A track crossing is up ahead, so you begin to speed up at 6.50 m/s2. How far do you drive before you pass the train?
You want the distances to be the same after t hours. So, noting that
6.5 m/s^2 = 23400 km/hr^2
105t = 40t + 23400/2 t^2
t = 1/180 hr
So, you overtake the train after 0.583 km
I hope you were not uncomfortable with the unusual units; you can always convert the speeds to m/s if you want. :-)
Well, I don't know about you, but I prefer to stay away from trains because they can be quite choo-choo-challenging to deal with. But let's calculate this out just for the fun of it!
First, we need to convert our speeds to meters per second (m/s). So, you were initially going 40.0 km/h, which is about 11.1 m/s. And the train was cruising at a steady 105 km/h, which is about 29.2 m/s.
Now, since the acceleration is given to us in m/s^2, we're all set! Let's use the formula of motion:
d = (v^2 - u^2) / (2a)
where u is the initial velocity, v is the final velocity, a is the acceleration, and d is the distance traveled.
Plugging in our values, we get:
d = (29.2^2 - 11.1^2) / (2 * 6.50)
Ignoring the math for a second, let's just say that you'd better buckle up, because you'll be driving approximately 830 meters before you pass that train! Keep your eyes on the road and stay safe.
To determine how far you drive before you pass the train, we can break down the problem into several steps:
Step 1: Convert the speeds to meters per second.
- Your speed: 40.0 km/h
To convert km/h to m/s, divide by 3.6:
40.0 km/h = (40.0 * 1000) / 3600 = 11.11 m/s
- Train's speed: 105 km/h
105 km/h = (105 * 1000) / 3600 = 29.17 m/s
Step 2: Determine the time it takes for you to catch up with the train.
Let's assume the time taken to catch up is t.
Step 3: Calculate the distance traveled by the train and by you during this time (t).
- Distance traveled by the train: train's speed * t
- Distance traveled by you: Initial speed * t + 0.5 * acceleration * t^2
Step 4: Set up an equation to solve for t.
According to the information given, the distances traveled by both you and the train during time t are the same. So we can set up the following equation:
train's speed * t = initial speed * t + 0.5 * acceleration * t^2
Step 5: Solve the equation for t.
Rearranging the equation, we get:
0.5 * acceleration * t^2 + (initial speed - train's speed) * t = 0
Using the quadratic formula, we can solve for t:
t = (-b +/- sqrt(b^2 - 4*a*c)) / (2*a)
Here, a = 0.5 * acceleration, b = (initial speed - train's speed), c = 0
Step 6: Calculate the distance traveled by you at time t.
Distance traveled by you = initial speed * t + 0.5 * acceleration * t^2
Now, let's plug in the values and calculate the distance.
Given:
Initial speed (your speed) = 11.11 m/s
Acceleration = 6.50 m/s^2
Train's speed = 29.17 m/s
Step 2: Determine the time it takes for you to catch up with the train.
Let's assume the time taken to catch up is t.
Step 5: Solve the equation for t.
a = 0.5 * 6.50 = 3.25
b = (11.11 - 29.17) = -18.06
c = 0
Using the quadratic formula:
t = (-(-18.06) +/- sqrt((-18.06)^2 - 4 * 3.25 * 0)) / (2 * 3.25)
t = (18.06 +/- sqrt(326.8836)) / 6.5
As we want the positive value of t, we can disregard the negative solution.
t = (18.06 + sqrt(326.8836)) / 6.5
Calculating the value of t, t ≈ 4.51 seconds.
Step 6: Calculate the distance traveled by you at time t.
Distance traveled by you = initial speed * t + 0.5 * acceleration * t^2
Distance traveled by you = 11.11 * 4.51 + 0.5 * 6.50 * (4.51)^2
Calculating the distance, we find that you drive approximately 96.85 meters before passing the train.
To find out how far you drive before you pass the train, we can set up a relative motion problem.
First, let's convert all the speeds to the same unit. Given that you are driving at 40.0 km/h and the train is moving at 105 km/h, we can convert these speeds to m/s:
Your speed = 40.0 km/h = (40.0 km/h) x (1000 m/1 km) x (1 h/3600 s) = 11.1 m/s (rounded to one decimal place).
Train's speed = 105 km/h = (105 km/h) x (1000 m/1 km) x (1 h/3600 s) = 29.2 m/s (rounded to one decimal place).
Now, let's calculate the time it takes for you to catch up to the train. We can calculate this by finding the time it takes for you to accelerate to the same speed as the train:
Using the kinematic equation: v = u + at, where:
v = final velocity (same as train's speed)
u = initial velocity (your speed)
a = acceleration
t = time
Let's plug in the values:
29.2 m/s = 11.1 m/s + (6.50 m/s^2) * t
Rearranging the equation:
t = (29.2 m/s - 11.1 m/s) / (6.50 m/s^2)
t = 18.1 m/s / 6.50 m/s^2
t = 2.79 seconds (rounded to two decimal places)
Now that we know the time it takes for you to catch up to the train, we can calculate the distance you travel during that time:
Using the equation: s = ut + (1/2)at^2, where:
s = distance
u = initial velocity (your speed)
t = time
a = acceleration
Plugging in the values:
s = (11.1 m/s)(2.79 s) + (1/2)(6.50 m/s^2)(2.79 s)^2
s = 30.9 m + 10.9 m
s = 41.8 meters (rounded to one decimal place)
Therefore, you will drive approximately 41.8 meters before you pass the train.