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A position function, often denoted as \(s(t)\), describes the location of an object along a coordinate line as a function of time \(t\). In simple terms, it tells you where the object is at any given time. This function is essential in physics and engineering to understand the motion of objects.
In the given exercise, the position function is defined as \(s(t)=1.86t^2\). This essentially means that the position of the body at any time \(t\), measured in seconds, is given by plugging \(t\) into this formula.
To find the position of the body at \(t=2\) seconds, you substitute \(t=2\) into the position function:

• Calculate \(s(2) = 1.86(2)^2 = 7.44\text{ feet}\).

Thus, at 2 seconds, the object is 7.44 feet along the specified line. This demonstrates the object's position on Mars during its free fall, measured at that precise moment.

The velocity function, denoted as \(v(t)\), describes how fast the position of an object is changing over time. In other words, it provides the rate of change of the position function \(s(t)\). Velocity not only tells us how fast an object is moving but also in which direction.
To find the velocity function from a position function, you differentiate \(s(t)\) with respect to \(t\). Differentiation involves calculating the derivative, which gives you the velocity function. Using the power rule in calculus, we differentiate \(s(t) = 1.86t^2\) to get:

• \(v(t) = \frac{d}{dt}(1.86t^2) = 1.86 \times 2t = 3.72t\).

Now, to find the velocity at \(t=2\) seconds, substitute \(2\) into \(v(t)\):

• \(v(2) = 3.72(2) = 7.44\text{ ft/s}\).

This means that at 2 seconds, the velocity of the body is 7.44 feet per second, indicating both the speed and the direction in which the body is moving during its free fall.

Speed is a scalar quantity that tells us how fast an object is moving regardless of its direction. It is the absolute value of the velocity. While velocity gives you both speed and direction, speed only measures how fast something is without indicating the direction.
To find the speed from the velocity function at a specific time, take the absolute value of the velocity. In the exercise, we calculated the velocity at \(t = 2\) seconds to be \(7.44\text{ ft/s}\). Since this value is positive, the speed at this moment is simply:

• \(|v(2)| = |7.44| = 7.44\text{ ft/s}\).

Thus, the speed of the body at 2 seconds is 7.44 feet per second. It's worth noting that if the velocity were to have been negative, the speed would still be the positive magnitude of that velocity. The concept of speed is critical in understanding motion as it provides a simple measure of the rate of motion, ignoring any directionality.