How to Calculate Distance Traveled: Methods & Sample Problems
Distance traveled describes how much path an object has covered in order to reach its destination in a specified time period. The distance covered formula for distance traveled is given as:
$d = vt$
Where,
$d$ = the distance traveled
$v$ = the velocity
$t$ = time taken to travel
Similar to the distance traveled formula, there is the distance traveled in the last second formula and formula for displacement which we will learn below.
Uses and Applications of Distance Traveled Formula
The distance-covered formula is applicable to compute the distance of driving a car or swimming stretch in a pool. While driving a car, the distance will be computed in kilometers or miles, the rate is in kilometers per hour or miles per hour, and time is in hours. While swimming laps in a pool, the distance is computed in laps.
Displacement in nth Second Formula
In order to calculate the displacement (position shift) from the velocity function, you just need to integrate the function. The negative areas below the x-axis subtracted from the total displacement. For this, we use a formula for displacement in nth second.
Displacement $= \int b_a v(t) dt$
In order to calculate the distance traveled we need to use the absolute value.
The displacement in the nth second distance formula covered in the nth second is given by $S_n = u + a(n - 12)$.
Derive the Expression for the Formula for Displacement in the nth Second
A is the acceleration
V is the velocity
U is the initial velocity
S is the distance
$s = ut + \dfrac{1}{2}at^2$
In order to calculate the distance traveled at the time of the nth second, we compute the distance covered in n seconds and subtract (minus-) the distance covered in (n-1) seconds and obtain:-
$s = un - u(n-1) + \dfrac{1}{2}an^2 - \dfrac{1}{2}a(n-1)^2$
Simplifying provides us
$u \text{ term }= u(n-(n-1)) = u \times 1 = u$
$(n-1)^2 = n^2-2n+1$
$a \text{ term }= \dfrac{1}{2}a(n^2-(n-1)^2) = \dfrac{1}{2}a(n^2-n^2+2n-1)$ the $n^2$'s cancel and provide us $\dfrac[1}{2a}(2n-1)$.
The final distance traveled equation for displacement in the nth second is
s = u + \dfra{1}{2a (2n-1)}$
Solved Examples
Example: A heavily loaded bus travels at a velocity of 80 miles per hour. Find out the total time taken by the truck to travel a distance of 300 miles?
Solution
Known: Velocity = 80 miles per hour
Displacement d = 300 miles
We know that,
Displacement $d = vt$
The time taken is provided by:-
$t = \dfrac{d}{v}$
$t = \dfrac{300}{80}$
$t = 3.75$ hours
Conclusion:
The distance formula is a method for calculating the distance between two places. These points can be of any size and in any dimension. Every day, people deal with the distance between two points. Inches, feet, miles, millimeters, and meters can all be used to describe it. The distance covered formula for distance traveled is given as: $d = vt$
FAQs on Distance Traveled Formula Explained
1. What is the fundamental formula for calculating distance traveled at a constant speed?
The most basic formula to calculate the distance an object travels is used when its speed is constant. The formula is: Distance = Speed × Time. For this formula to be accurate, the object must not be accelerating or decelerating; it must maintain the same speed over the entire duration of the time period.
2. How do you find the distance traveled when an object is under uniform acceleration?
When an object moves with a constant or uniform acceleration, you must use the equations of motion. The primary formula for distance is:
s = ut + (1/2)at²
Where:
- s is the distance traveled
- u is the initial velocity of the object
- t is the time for which the object was in motion
- a is the uniform acceleration
3. What is the key difference between distance traveled and displacement?
The primary difference lies in their definitions as physical quantities:
- Distance is a scalar quantity that represents the total length of the path covered by an object. For example, if you walk 10 metres north and then 10 metres south, your distance traveled is 20 metres.
- Displacement is a vector quantity that represents the shortest straight-line path between the object's starting and ending points. In the same example, since you ended up where you started, your displacement is zero.
4. How can you calculate the distance traveled if the time taken is unknown?
If the time of travel (t) is not given, but you know the initial velocity (u), final velocity (v), and uniform acceleration (a), you can use the third equation of motion to find the distance (s). The formula is:
v² = u² + 2as
By rearranging this formula, you can solve for distance: s = (v² - u²) / 2a. This is extremely useful for problems where time is not a factor.
5. Why can't the simple formula 'Distance = Speed × Time' be used for an accelerating object?
The formula Distance = Speed × Time is only valid when the speed is constant. For an accelerating object, the speed is continuously changing every moment. If you were to use this formula, you would not know which value of speed to use—the initial speed, the final speed, or an average? The equations of motion, like s = ut + (1/2)at², are designed specifically to integrate this continuous change in speed over time to give an accurate value for the distance traveled.
6. What is the formula for the distance traveled in the nth second of motion?
To find the distance covered during a specific second of motion (for example, during the 5th second), you use a special formula derived from the equations of motion:
Sₙ = u + (a/2)(2n - 1)
Here, Sₙ is the distance traveled during the nth second, u is the initial velocity, and a is the uniform acceleration. This formula calculates the displacement for that one-second interval only, not the total distance from the start.
7. How is distance traveled calculated using calculus for advanced problems?
For students in higher grades or those dealing with non-uniform acceleration, calculus provides the most accurate method. If the velocity of an object is expressed as a function of time, v(t), the distance traveled (s) between two time points, t₁ and t₂, is the definite integral of the velocity function:
s = ∫ (from t₁ to t₂) v(t) dt
This method calculates the total area under the velocity-time graph, which precisely represents the distance traveled, even when acceleration is not constant.
