The velocity ( mi \operatorname{mi} mi / h r hr ) of a drone flying in the air is given by v ( t ) = 12 + 4 t 2 v\left(t\right)=12+4t^2 v ( t ) = 12 + 4 t 2 for 0 ≤ t ≤ 4 0\le t\le4 0 ≤ t ≤ 4 hours. Let s ( 0 ) = 0 s\left(0\right)=0 s ( 0 ) = 0 . How far has the drone traveled by the time it has reached 48 mi 48\operatorname{mi} / h r hr ?

we've been spending some time working through this problem where we are given a drone. We are given the velocity function of that drone in miles per hour, and we are also given this time interval right here. And we were initially asked to find position. Now we actually found the position s of t. We were we figured out this function a couple videos ago, and we've been working through this problem where we're now asked to figure out how far the drone has traveled by the time it has reached 48 miles per hour. Now this is a bit of a tricky problem because you might notice here that we're given 48 miles per hour here, which is a velocity or a speed. And notice this is our position function right here. So it's not like we can just directly plug something into this function to get our answer. So the way that we actually need to do this is by going to our velocity function. Now how exactly could we figure out how far the drone has traveled? Well, first, what we need to do is figure out a time. And the way we can do this is by taking our velocity function, v of t, and recognizing this is 12 plus four t squared. And we're trying to first figure out what time we're going to reach 48 miles per hour. Well, to do this, I can just take my velocity function instead of equal to 48 miles per hour. So we're going to have that 12 plus four t squared is equal to 48 because that's everything here. And since our velocity function is representing miles per hour for our units, then we can use all the same numbers here. So doing this, what I'll first do is subtract 12 on both sides of this equation. Now we get the twelves to cancel there, leaving me with four t squared is equal to 36. Now I can divide both sides of this equation by four, getting the fours to cancel there, giving me that t squared is equal to 36 divided by four, which is nine. And from here I can take the square root on both sides of the equation to get that t is equal to plus or minus the square root of nine, which is three. So this right here would be our time, t. Now notice that we get both a positive and negative result, but we're also given a time interval from zero to four. Negative three is going to fall outside of this time interval that we're given here, so we only wanna focus on the positive value of three. So we only care about three being the time that we look at. So if t is equal to three, then what we need to do is go back up to our position function that we already found and plug in three. So we're going to have four times three cubed over three plus 12 times three. Now from here I can recognize that one of the threes will cancel up here so it's going to give me four times three squared plus 12 times three and all of this here should come out to 48 and it's this is a position it's going to be 48 miles. So s of three is equal to 48 miles, and that is the solution to this problem. So as you can see, it took a bit of work since we first needed to figure out this time. Once we had that, we could just use our same position function in solving this problem. It's just like how we solve other problems when dealing with position. So hope you found this video helpful, and let's move on.

10. Physics Applications of Integrals

Kinematics

Calculus

10. Physics Applications of Integrals

Kinematics

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Multiple Choice

The velocity ( $\operatorname{mi}$ / $h r$ ) of a drone flying in the air is given by $v\left(t\right)=12+4t^2$ for $0\le t\le4$ hours. Let $s\left(0\right)=0$ .
How far has the drone traveled by the time it has reached $48 mi$ / $h r$ ?

24 mi

34.7 mi

48 mi

72 mi

Verified step by step guidance

Step 1: Understand the problem. The velocity of the drone is given as v(t) = 12 + 4t^2 (in mi/hr), and we are tasked with finding the distance traveled by the drone when its velocity reaches 48 mi/hr. The initial position is s(0) = 0.

Step 2: Solve for the time t when the velocity v(t) equals 48 mi/hr. Set v(t) = 48 and solve the equation 12 + 4t^2 = 48 for t.

Step 3: Once t is determined, use the formula for displacement, which is the integral of velocity over time. The displacement s(t) is given by s(t) = ∫ v(t) dt. Substitute v(t) = 12 + 4t^2 into the integral.

Step 4: Evaluate the definite integral of v(t) from t = 0 to the value of t found in Step 2. This will give the total distance traveled by the drone.

Step 5: Add the initial condition s(0) = 0 to the result of the integral to find the total distance traveled. This will give the final answer.

1:17m

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The velocity (mi\operatorname{mi}mi/hrhr) of a drone flying in the air is given by v(t)=12+4t2v\left(t\right)=12+4t^2v(t)=12+4t2 for 0≤t≤40\le t\le40≤t≤4 hours. Let s(0)=0s\left(0\right)=0s(0)=0.How far has the drone traveled by the time it has reached 48mi48\operatorname{mi}/hrhr?

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The velocity ( $\operatorname{mi}$ / $h r$ ) of a drone flying in the air is given by $v\left(t\right)=12+4t^2$ for $0\le t\le4$ hours. Let $s\left(0\right)=0$ .
How far has the drone traveled by the time it has reached $48 mi$ / $h r$ ?