Total distance traveled question

In summary: When you integrate from 0 to 5, you are calculating the total distance traveled in the positive direction. But since the particle can also move in the negative direction, you need to subtract the distance traveled in the negative direction (from 4 to 5) in order to get the total distance traveled. So, the formula for total distance traveled would be the integral of the absolute value of v(t) from 0 to 5. In summary, when calculating total distance traveled using the velocity equation for a particle, you need to integrate the absolute value of v(t) from the starting point to the ending point, taking into account both positive and negative distances. This results in subtracting the distance traveled in the negative direction from the total
  • #1
Marioqwe
68
4

Homework Statement



Lets say we have the velocity equation for a particle

v(t) = at^3 - bt^2 + ct^ - d with t between 0 and 5

So, to find its displacement I have to integrate v(t) from 0 to 5, and I understand why.
But if I want to find the total distance traveled, I must find where t is negative and then i integrate according to that. So, I would have something like an integral of v(t) from 0 to 4 minus an integral of v(t) from 4 to 5. But why do we subtract? Can anybody explain that to me? We also consider moving backwards when we calculate the total distance traveled right?

Thanks
 
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  • #2
Marioqwe said:

Homework Statement



Lets say we have the velocity equation for a particle

v(t) = at^3 - bt^2 + ct^ - d with t between 0 and 5

So, to find its displacement I have to integrate v(t) from 0 to 5, and I understand why.
But if I want to find the total distance traveled, I must find where t is negative and then i integrate according to that.
No, your interval for t is [0, 5], so t is never negative. You need to find where v(t) is negative, because that's when the particle is moving backwards.

If you integrate v(t) from 0 to 5 you'll get the displacement, which is the distance between the particle's position at time t = 0 and time t = 5. For example, if the particle started at the origin, then moved right 5 units, then back 7 units, its displacement would be 2 units, but the distance traveled would be 12 units.
Marioqwe said:
So, I would have something like an integral of v(t) from 0 to 4 minus an integral of v(t) from 4 to 5. But why do we subtract? Can anybody explain that to me? We also consider moving backwards when we calculate the total distance traveled right?
Yes.
 

1. What is the formula for calculating total distance traveled?

The formula for calculating total distance traveled is distance = speed x time. This formula is based on the basic formula of distance = rate x time, where rate is equivalent to speed.

2. Can total distance traveled be negative?

No, total distance traveled cannot be negative. Distance is a scalar quantity, meaning it only has magnitude and no direction. Therefore, it is always positive or zero.

3. How do you calculate total distance traveled when there are multiple segments?

To calculate total distance traveled when there are multiple segments, you need to calculate the distance traveled for each segment and then add them together. For example, if a car travels 50 km north and then 30 km east, the total distance traveled would be 80 km.

4. Can total distance traveled be greater than the actual distance between two points?

No, total distance traveled cannot be greater than the actual distance between two points. This is because distance is a measure of the length of the path taken, not the displacement or straight-line distance between two points.

5. How does the total distance traveled affect the overall journey?

The total distance traveled can affect the overall journey in terms of time, fuel consumption, and wear and tear on the vehicle. A longer total distance traveled would typically result in a longer journey time and higher fuel consumption. It may also put more strain on the vehicle, potentially leading to more maintenance and repairs.

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