The position (in meters) of a marble, given an initial veloc...

Question

The position (in meters) of a marble, given an initial velocity and rolling up a long incline, is given by s = 100t / t+1, where t is measured in seconds and s=0 is the starting point.

b. Find the velocity function for the marble.

Accepted Answer

Welcome back everyone to another video. A particle moves along a line such that its position is given by the equation P of T equals 50T divided by T + 2, where P is the position in meters from the origin and T is time in seconds. Calculate the particle's velocity function. We're given 4 answer choices A says -100 divided by T + 2 squad, B 100 divided by T + 2, C 10 divided by T + 2 squad, and D 100 divided by T + 2 squad. So in this problem we're given the position function P of T is given as 50 T. Divided by T + 2, we are looking for the velocity function, and we can recall that the velocity function can be found by finding the derivative of the position function. So now we understand the goal of the problem. We want to identify the first derivative of the position function, so we're going to substitute the expression of the position function 50T divided by T + 2, and we want to differentiate it. So now here we have a rational function, and if we want to find its derivative, we want to apply the quotient rule which says that first of all we take the derivative of the top function, so we are differentiating 50 T. Multiplying that by the bottom function, which is T + 2, then we are subtracting the first function, which is 5 TT. Multiplied by the derivative of the bottom function T + 2. And then in the denominator, we want to square the bottom function which gives us T + 2 squared. And from here our goal is to perform intermediate steps and evaluate each derivative. So 50, well, the derivative of this term can be found by differentiating T because 50 is a constant and we can factor it out, and the derivative of T is 1, right, because it's simply dt divided by D T. So it's 50 multiplied by 1, which is 50, and for the derivative of T plus 2, well, once again we have the derivative of T, which is 1, and the derivative of 2 is 0, right? So now that we have our derivatives, we can see that our numerator becomes 50, multiplied by T + 2. Then we are subtracting 50T, right? And we are multiplying it by 1. So it's just minus 50 T in the numerator. In the denominator, we still have T + 2 squared, and from here we can just factor out 50. Which gives us T + 2 minus T. All of that divided by T + 2 squared. And we can see that T minus T gives us 0, so we end up with 15 multiplied by 2, which is 100 divided by. T + 2 squad, which is our velocity function, and as we can see it corresponds to the answer choice d. Thank you for watching.

The position (in meters) of a marble, given an initial velocity and rolling up a long incline, is given by s = 100t / t+1, where t is measured in seconds and s=0 is the starting point.
b. Find the velocity function for the marble.

Key Concepts

Position Function

Velocity Function

Differentiation

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