A train passing at point A at a speed of 72 kph accelerates at .75 m/s2 for one minute along a straight path then decelerated at 1.0 m/s2. How far in km from point A will it be 2 minutes after passing point A?

First of all, make the conversion

72 km/h = 20 m/s

After accelerating 60 seconds at 0.75 m/s^2, the speed increases by 45 m/s to 65 m/s. The average speed during that minute is 105/2 = 52.5 m/s, and the distance travelled is 52.5 m/s * 60 s = 3150 m.

After one more minute (60 seconds) of decelerating at -1.0 m/s^2, the speed drops to
105 - 60 = 45 m/s. The average speed during the second minute is (105 + 45)/2 = 75 m/s. The distance traveled during the second minute is 75*60 = 4500 m.

Add the distances traveled during the first and second minutes for the final answer.
You could also integrate the V(t) function for two minutes. The answer should be the same.

Well, well, well, it seems we have a speeding train with a bit of a personality! Let's crunch some numbers and find out where it'll end up, shall we?

First things first, let's convert that pesky acceleration from m/s² to km/h². To do that, we need to multiply it by 129.6 ('cause there are 129.6 km in an hour, you know).

So, the initial acceleration of 0.75 m/s² becomes a whopping 97.2 km/h², all ready to make us dizzy!

Since the train accelerates for one minute, we can simply add that initial speed of 72 km/h to the acceleration to find the final speed after one minute.

So, 72 km/h + 97.2 km/h² * (1/60 h) = 73.62 km/h.

Now, over the next minute, the train is going to decelerate at a rate of 1.0 m/s², which is 129.6 km/h².

Since we're dealing with a deceleration, we need to subtract that acceleration from the current speed to see how fast the train will be going two minutes after point A.

So, 73.62 km/h - 129.6 km/h² * (1/60 h) = -56.98 km/h! Yikes, that's one slow train!

Now, to find out how far the train has traveled, we use the average of the initial and final speeds and multiply it by the time.

(72 km/h + (-56.98 km/h)) * 2 min = 30.04 km.

Therefore, the train will be approximately 30.04 km away from point A after two minutes.

But remember, even though it's a slow train, it's also a funny train! So, don't take it too seriously!

To find the distance traveled by the train 2 minutes after passing point A, we need to break down the motion into different phases.

Phase 1: Acceleration for 1 minute
The initial speed of the train is 72 kph, and it accelerates at a rate of 0.75 m/s^2 for 1 minute. Let's calculate the initial speed in m/s:

72 kph = 72 * (1000/3600) m/s = 20 m/s

Using the equation of motion s = ut + (1/2)at^2, where s is the displacement, u is the initial velocity, a is the acceleration, and t is the time, we can find the distance covered during acceleration:

s1 = (20 * 60) + (0.5 * 0.75 * 60^2)
= 1200 + 1350
= 2550 m

Phase 2: Constant speed for 1 minute
After the acceleration phase, the train maintains a constant speed. The distance covered during this phase can be calculated using the formula s = vt, where v is the constant velocity and t is the time:

s2 = 20 * 60
= 1200 m

Phase 3: Deceleration for 1 minute
Now, the train decelerates at a rate of 1.0 m/s^2 for 1 minute. The final velocity after deceleration can be calculated using the equation v = u + at:

Final velocity = 20 - (1.0 * 60)
= 20 - 60
= -40 m/s

The negative sign indicates the direction of velocity change (deceleration). To find the distance covered during this phase, we use the formula s = ut + (1/2)at^2:

s3 = (20 * 60) + (0.5 * -1.0 * 60^2)
= 1200 - 1800
= -600 m (The negative sign indicates the direction)

Total distance traveled after 2 minutes = s1 + s2 + s3
= 2550 + 1200 - 600
= 3150 m

Finally, converting the distance to kilometers:
Total distance traveled after 2 minutes = 3150 m / 1000
= 3.15 km

Therefore, the train will be 3.15 km from point A, 2 minutes after passing it.

To find the distance traveled by the train 2 minutes after passing point A, we need to calculate the distance traveled during acceleration, the distance traveled during deceleration, and the distance traveled during constant speed.

Let's break down the problem step by step:

Step 1: Calculate the distance traveled during acceleration.
First, we need to find the time the train spends accelerating. Given that the train accelerates for one minute, the time taken for acceleration (t1) is 1 minute.

Next, we can use the formula to calculate the distance (d1) traveled during acceleration:
d1 = (1/2) * acceleration * (time)^2

Plugging in the values we have:
acceleration = 0.75 m/s^2
time = 1 minute = 60 seconds

Converting the distance from meters to kilometers:
1 meter = 0.001 kilometers
So, d1 (in kilometers) = (1/2) * 0.75 * (60)^2 * 0.001

Step 2: Calculate the distance traveled during deceleration.
Similar to the calculation for acceleration, we need to find the time spent decelerating. Given that the total time after passing point A is 2 minutes, and the acceleration time is 1 minute, the time taken for deceleration (t2) is 2 - 1 = 1 minute.

Using the same formula, we can calculate the distance (d2) traveled during deceleration:
d2 = (1/2) * deceleration * (time)^2

Plugging in the values we have:
deceleration = -1.0 m/s^2 (considering the negative value for deceleration)
time = 1 minute = 60 seconds

Converting the distance to kilometers:
d2 (in kilometers) = (1/2) * (-1.0) * (60)^2 * 0.001

Step 3: Calculate the distance traveled during constant speed.
To find the distance traveled during constant speed, we need to calculate the distance covered in the remaining time, which is 2 minutes - (acceleration time + deceleration time) = 2 - 1 - 1 = 0 minutes.

During constant speed, the formula to calculate distance (d3) is:
d3 = speed * time

Plugging in the values we have:
speed = 72 kph = 72/60 = 1.2 km/min (converting from kph to km/min)
time = 0 minutes

Therefore, d3 = 1.2 * 0 = 0 kilometers.

Step 4: Total distance traveled.
Now, we can add up the distances traveled during acceleration, deceleration, and constant speed to get the total distance traveled by the train.
Total distance = d1 + d2 + d3

Calculate and sum up the values obtained in Steps 1, 2, and 3 to find the total distance traveled by the train.

Finally, the total distance traveled by the train in kilometers 2 minutes after passing point A is equal to the sum of the distances calculated in the previous steps.