Acceleration to position Given the following acceleration func...

Question

Acceleration to position Given the following acceleration functions of an object moving along a line, find the position function with the given initial velocity and position.

a(t) = -32; v(0) = 20, s(0) = 0

Accepted Answer

Welcome back, everyone. Given the acceleration function A of T equals -24, with initial conditions V of 0 equals 15 and S of 0 equals 5, find the position function S of T. So first of all, let's recall that. The velocity function V of T is the integral of the acceleration function A of TDT and also our position function as of T is going to be equal to the integral of the velocity function V of TDT. So we can begin by integrating our acceleration function to identify V of T, right? We can state that V of T, our velocity function. Is the integral of 24 DT. We can factor out the constant, that's -24 and the integral of the T is just T, right? We add a constant of integration C1, and then we can identify C1 from the initial condition V of 0 equals 15. Which essentially means that V is 15 when C is 0. So we set 15 equals -24 multiplied by 0. Plus C1. So 15 equals 0 plus C1, meaning C1 equals 15. And now we have shown that our velocity function is -204 T + 15. From here finally we can solve for the position function as of T, which is the integral of the velocity function that we have just found. So the integral of -204 T + 15 dT. Let's evaluate each integral, right? We have a sum, so we can begin with -24. Let's factor out the constant. The integral of T is T2 divided by T using the power rule. And then we add 15, the integral of the C is just T, right? Plus a constant of integration C2, because once again we have another indefinite integral. And now our goal is to identify C2 from the second condition. If T is 0 S is 5, so we set 5 equals. 24 divided by 2 is -12, right? So -12 multiplied by 0 squared. Plus 15 multiplied by 0. Plus C2. So from here we have 5 equals C2 or C2 equals 5. We can also simplify -24 divided by 2 is -12 and our position function, and our final step is to substitute C2 equals 5. So the complete position function as of T becomes -12 T squad. Plus 15 T + C2, so plus 5. That would be our final answer for this problem, and thank you for watching.

Acceleration to position Given the following acceleration functions of an object moving along a line, find the position function with the given initial velocity and position.

a(t) = -32; v(0) = 20, s(0) = 0

Key Concepts

Acceleration

Velocity

Position Function

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