What Is the Toal Distance Traveled by the Particle During First 3 Seconds

Watch the video for two examples of how to find full distance traveled in calculus—on an interval [0, 5]— or read on below:

How To Discover Full Distance on an interval [0. 5]

Can't see the video? Click hither.

Contents (Click to skip to that section):

1. How to Detect Total Distance
   • With Derivatives
   • With Integrals
2. Distance From a Point To a Line
3. Total Altitude vs. Total Deportation
4. How to Find Total Displacement

How to Find Total Distance

Most distance problems in calculus requite you lot the velocity function, which is the derivative of the position function. The velocity formula is ordinarily presented as a quadratic equation.

You can observe total altitude in two different means: with derivatives, or by integrating the velocity function over the given interval.

How to Discover Full Distance with Derivatives

Example problem: Find the total distance traveled for a particle traveling in a horizontal motion from t = 0 to t = five seconds co-ordinate to the position function:
s(t) = 8t² – 4t.

Step 1: Find the velocity role. The velocity function is the derivative of the position function. So, to detect the velocity part y'all demand to differentiate south(t) = 4t² – 2t, using the power rule and derivative of a constant rule:
v(t) = 16t – 4

Step two: Fix the velocity part to nothing, then solve, to find where the velocity function changes direction:

1. five(t) = 16t – 4 = 0
2. 16t – four (+ four) = 0 (+ 4)
3. 16t(/sixteen) = 4 (/xvi)
4. t = 4/xvi = 1/4

Notation: Alternatively, you could graph your velocity role and notation where areas of the graph are above or below the x-axis. This where the velocity as a vector has changed direction.

Step 3: Solve the position function for the starting and ending interval values (which are 0 and 5, according to the question), and any t-values yous found in step ii:

• s(0) = 8(0)² – 4(0) = 0.
• s(i/4) = eight(¼)^two – 4(1/4) = -½
• southward(5) = 8(five)² – 4(5) = 180.

This gives you the position of the object at each time interval.

Pace iv: Notice the distance traveled between each betoken. These are vectors, and so we have to utilize absolute values to detect the distance:

• Between 0 and -½, distance = |-½ – 0| = ½
• Between -½ and 180, distance = |180 – (-½)| = 180½

Step 5: Add your values from Stride 4 together to discover the total distance traveled.
½ + 180 ½ = 181

How to Find Total Altitude: Integrals

Example question: A particle travels according to the post-obit velocity part:
v(t) = 3tⁱⁱ – 12.
What is the total distance traveled from t = 1 to t = 3?

Stride 1: Set the velocity equal to cypher to observe where the function changes management:

• v(t) = 3tⁱⁱ – 12 = 0
• 3(t² – 4) = 0 (factoring out).
• three(t – 2)(t + two) = 0
• t = -2, +2

We're working with the airtight interval [one, three] (from the question), and so the point we're interested in (the i that falls in the interval) is t = 2.

Step two: Integrate the velocity function for each interval. To solve, you need two dissever definite integrals:

Step three: Add the absolute values of the amounts you calculated in Footstep ii:
v + 7 = 12.
The total distance traveled is 12 units.

Calculating the Altitude From a Point To a Line

The distance from a bespeak to a line is the length of the shortest path between that point and the place on the line nearest to it. This path is always the line that is perpendicular (at correct angles) to your original line.

1. If y'all know the coordinates

If you know the Cartesian coordinates of your point and y'all take an equation for your line, yous can calculate the distance between the bespeak and the line with the formula:

Where:

• 10₀, y₀ are the coordinates of the point,
• a, b, and c are the coefficients (and abiding) for the line a x + b y + c = 0.

Example

Suppose your line was given by 4 ten + y = 0, and your point was (2, ii).

Every bit you take coordinates, y'all can utilize the kickoff equation given higher up.

The distance from the point to the line would be:
|4 · 2 + 1 · 2 | / √(4+ 4).
This is equivalent to |10|/(2 √2), or approximately 3.54.

2. If yous don't have coordinates

Sometimes you don't take an equation for the line. If all yous know most your line is two points that it crosses, you can apply another formula to summate that altitude between a betoken and the line. Allow's make our point ten₀, y₀ again, and ascertain the point as i which goes through 10₁, y₁ and ten_two, y_two . So the altitude is given by

.

These formulas can be proved through algebra, through geometry,and through vector analysis.

Total Distance vs. Total Displacement

The definite integral of the velocity role of an object gives yous the full deportation—how far an object is from a indicate of origin. However, this is different from distance traveled. Let's say you were going to the local market, which is about 1 mile from your firm. The total displacement, when yous reach your destination, is ane mile. When you get dwelling, your total displacement is zero (you're back at your starting bespeak). Just the distance traveled is 2 miles: one mile to the market, and i mile back once more. Displacement doesn't reflect distance traveled.

If an object is traveling in one direction (east.g. along the x-axis), without reversing or changing class, and then total distance is the same equally total displacement.

How to Find Total Displacement

The formula for full displacement is:

Where:

• Σ = summation notation (add everything up)
• Δx_i are the individual displacements.

Instance question: A automobile travels 2 miles Eastward then 4 miles W. What is the total deportation?

Solution:

At this phase, information technology may assistance to draw a graph of the problem. Here's what nosotros are trying to find in this question:
The moving-picture show shows that the car is -ii miles from its starting position. We can testify that mathematically with the formula.

Footstep 1: Calculate the individual displacements (Δx_i) using the displacement formula:
Δx = x_f – x₀
Where:

• x_f = final position,
• ten₀ = starting position.

For this question nosotros have two individual displacements: 2 miles E and 4 miles W.

• two miles Eastward: We started at position "0" and concluded at "2", and so:
  Δx = 2 – 0 = 2
• two miles S: Nosotros started at "ii" and ended at -ii, and so:
  Δx = -2 – (2) = -4

Stride 2: Add up the private displacements yous calculated in Step one:

two + (-4) = -ii.

That's information technology!

References

Gore, Bhalchandra. On Finding the Shortest Distance from a Indicate to a Line. Retrieved from
https://www.ias.ac.in/article/fulltext/reso/022/07/0705-0714 on December xv, 2018.
Distances. Math 21a, Fall 2008. Retrieved from http://www.math.harvard.edu/archive/21a_fall_08/handouts/distances/distance.pdf on Dec 15, 2018.
Larson, R. (2011). Calculus 1 with Precalculus. Cengage Learning; three edition.

---------------------------------------------------------------------------

Demand help with a homework or test question? With Chegg Written report, you can become step-by-stride solutions to your questions from an expert in the field. Your kickoff 30 minutes with a Chegg tutor is complimentary!

Share this post