Acceleration A drag racer accelerates at a(t)=88 ft/s². Assume...

Question

Acceleration A drag racer accelerates at a(t)=88 ft/s². Assume v(0)=0, s(0)=0, and t is measured in seconds.

c. At this rate, how long will it take the racer to travel 1/4 mi?

Accepted Answer

Welcome back, everyone. A cyclist starts from rest and accelerates at a constant rate of A of T equals 32 ft per second squared. With V of 0 equals 0 and S of 0 equals 0, and T in seconds. How long does it take to cover 400 yards? 1 yard is equal to 3 ft. So for this problem, we're going to identify our position function first of all, right? We're going to begin with the acceleration function. We know that A of T is equal to 32 ft per second squared, and we can identify our velocity function B of T by integrating the acceleration function with respect to time. So let's identify the integral of 32 DT, which is 32 T and we are going to add a constant of integration C1. We are going to identify C1 knowing that V of 0 equals 0. This means that if T is 0, then velocity is 0 as well. So we substitute 0 for velocity, 0 for time. And we can show that our constant C1 is equal to 0. So we have our velocity function V of T equals 302 T. And then we can identify the position function as of T by integrating the velocity function. With respect to time, so that would be the integral of 30 to TDT. We got 32 multiplied by T2d divided by 2, using the power rule, and we are adding a constant of integration C2. Let's simplify this as 16 T2 plus C2. And now we can use the initial condition as of 0 equals 0. So if T is 0, S is 0. We sets equals 0. And C equals 0. So we got 0 equals. 16 multiplied by 0 squared plus C2. So C2 must be 0. Well then, so we have our position function. As of T equals 16 T squared. Now, let's understand that time is greater than or equal to 0, so our position function is always positive. So what we're going to do is simply consider S of T equals 400 yards, and because 1 yard is 3 ft, right, we are going to have a total distance of 1200 ft, 3 multiplied by 400. So we set S of T equals 1200. That's because S of T. Is measured in feet right since we started with the acceleration function of feet per second squared. That's why we needed to perform this conversion. And now since S of T is 16 t squad, we set it equal to 1200. We can divide both sides by 16, so T squared is 1200 divided by 16. And now we can solve our time by taking the positive square root off. 1200 divided by 16. Now let's simplify. Square root of 16 is 4, so we got 4 in the denominator, entering our numerator, we can rewrite 1200 as 400 multiplied by 3, so we get 20 square root of 3. And now 20 divided by 4 is 5, so we get 5 square roots of 3 seconds, which is our final answer for this problem. Thank you for watching.

Acceleration A drag racer accelerates at a(t)=88 ft/s². Assume v(0)=0, s(0)=0, and t is measured in seconds.

c. At this rate, how long will it take the racer to travel 1/4 mi?

Key Concepts

Acceleration and Its Relationship to Velocity and Displacement

Integration of Constant Acceleration

Unit Conversion and Distance Measurement

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