A turtle crawls along a straight line, which we will call the x-axis with the positive direction to the right. The equation for the turtle's position as a function of time is x(t) = 50.0 cm + (2.00 cm/s)t − (0.0625 cm/s2)t2. (e) Sketch graphs of x versus t, υx versus t, and ax versus t, for the time interval t = 0 to t = 40 s.

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Hey, everyone in this problem, a toy travels in a long straight line following the equation X of T is equal to 40 m plus four m per second, T minus 0.485 m per second squared T squares, which describes the toy's position as a function of time. We're told to take the motion to be along the X axis and the positive X direction to be to the right. We're asked to draw a graph of X versus T for the interval from 0 to 50 seconds. We're given five answer choices. Option A, it shows a graph of X versus T X as in meters time is in seconds. The function startss at zero seconds at about 40 m. It increases quadratic to a maximum of about 125 m at around 40 seconds and then slightly decreases When it hits that 52nd mark. Option B, this also looks quadratic OK. But this time it starts at a position of X equals zero m and increases quadratic option C and this also looks quadratic, It starts at 0m at zero seconds and it's going to decrease quadratic. It reaches a minimum around 40 seconds. The minimum value is around and then slightly increases to the 52nd mark. Option D shows a linear relationship. OK. We have a negative slope starting at X equals zero m and decreasing down to X equals negative 20 m at seconds. And option E is also a linear relationship starting at x equals 40 m And decreasing down to x equals 20 m at 50 seconds. Now, right away, we can go ahead and eliminate two answer choices. We are given the function or the equation of the motion X as a function of time. And it's a quadratic equation. OK? There's a T squared term, A T term and a constant term. So we know this is going to be quadratic, it's going to look like part of a parabola. So we can eliminate option E and option D because they both show linear relationships, which is not what we have. OK? Going from there. And we have our function X of T Is equal to m Plus four m/s, multiplied by T -0.04, meters per second squared T squared. Now, we're gonna start by finding the Y intercept because the answer choices we have have a few different Y intercepts. If we look at our Y intercept, Hey, this is gonna be one T Is equal to zero seconds and when T is equal to zero seconds, we have that X of zero is equal to 40 m plus zero zero, which is just equal to 40 m. So we know that at T equals zero seconds, our graph should show a position of 40 m looking at our answer choices. OK. That's already enough to figure this problem out. Option B and option C both show quadratic functions or quadratic shapes That start at x equals zero m. That means that the graph matching our equation is going to be option a, we have this quadratic function starting at 40 m. Now, if we wanted to go further and double check some of these details, we can't and we could find the X intercepts. Next, the X intercepts occur when the position X is equal to zero. OK. So we can substitute that, that zero, we can find the two T values. OK. That'll, that will give us two T values like the roots of this equation. You can find those intercepts and compare that with what we have on the graph. And you can also find the value of X at 50 seconds. And if you substitute 50 seconds into the equation for T You're gonna get around 118.75 m which also matches with that end value at T equals 50 seconds in graph a. So the correct answer here is option A that matches with all those details that we found. Thanks everyone for watching. I hope this video helped see you in the next one.

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Key Concepts

Position Function

Velocity

Acceleration