Day hike The velocity (in miles/hour) of a hiker walking...

Day hike The velocity (in miles/hour) of a hiker walking along a straight trail is given by $v(t)=3 \sin ^{2}(\pi t / 2),$ for $0 \leq t \leq 4$ Assume that $s(0)=0.$ a. Determine and graph the position function, for $0 \leq t \leq 4.$ b. What is the distance traveled by the hiker in the first 15 min of the hike? (Hint: $\sin ^{2} t=\frac{1}{2}(1-\cos 2 t).$ c. What is the hiker's position at $t=3 ?$

Day hike The velocity (in miles/hour) of a hiker walking along a straight trail is given by $v(t)=3 \sin ^{2}(\pi t / 2),$ for $0 \leq t \leq 4$ Assume that $s(0)=0.$
a. Determine and graph the position function, for $0 \leq t \leq 4.$
b. What is the distance traveled by the hiker in the first 15 min of the hike? (Hint: $\sin ^{2} t=\frac{1}{2}(1-\cos 2 t).$
c. What is the hiker's position at $t=3 ?$

Key Concepts

Graphical Representation of Motion

Graphing the position function over a specified interval provides a visual interpretation of the motion. Through this graphical analysis, one can identify key aspects such as starting points, changes in speed, and overall displacement, which are critical for understanding the dynamics of the moving object.

Definite Integration for Distance Calculation

Definite integration is used to compute the exact change in position over a given time interval. In problems involving distance, especially when differentiating between displacement and distance traveled, the integration of the absolute value of velocity (or speed) over the interval gives the total distance the object has moved.

Trigonometric Identity for Simplifying Integration

Trigonometric identities, like converting sin²t into ½(1 - cos2t), are powerful tools in simplifying the integration process. By applying these identities, complex trigonometric expressions become more manageable, allowing for straightforward integration using standard techniques.

Antiderivatives and Initial Conditions

When determining the position function through integration, an arbitrary constant (the constant of integration) is introduced. Initial conditions, such as an initial position at time zero, are then used to evaluate this constant, ensuring that the position function accurately reflects the starting point of the motion.

Integration for Position Determination

In kinematics, the position function is obtained by integrating the velocity function over time. This concept relies on the fundamental theorem of calculus, which connects the derivative (velocity) and the antiderivative (position). Integrating the velocity function produces a position function whose derivative returns the original velocity, making this method essential for analyzing motion.

Transcript

00:01 Okay, so in this problem, we have a hiker who is traveling on a straight trail, and his velocity is determined by the equation function we have here.

00:12 Velocity with respect to time is equal to three times sine squared of pi t over two.

00:19 And we know that at distance, our time equals zero, the distance is also equal to zero.

00:24 So what we're going to do first is we're going to figure out the position equation, or position function for this.

00:29 So the first thing we need to understand for this is that we need to know that sine squared t is equal to one -half times one minus cosine 2t.

00:48 If we translate that to the equation that we have at hand, we figure out that in our case, we have s of t is equal to 3 over 2 times.

01:15 1 minus oh sorry velocity of cosine times 2 pi t over 2 the 2's cancel and what we can do from here is we're gonna and do the antiderivit of this to figure out s of t so to find s of t this is going to become 3 over 2 the 1 is going to become a t the cosine is to become a sign and because we have a pi being multiplied by the t within the signs parameters or dividing by pi.

01:56 So s of t is equal to this.

01:59 If you were to graph it, it would look something like this.

02:06 Okay, and we have a bound of four.

02:10 Then for b, for the next part, it's asking us to figure out the distance that our hiker has walked in 15 minutes.

02:22 So note, this is in miles per hour...

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