91Ó°ÊÓ

To understand how to find velocity and acceleration from a position function, it's essential to grasp the chain rule. The chain rule is a fundamental property for differentiating composite functions. When functions are nested inside each other, like in our example, you must differentiate the outer function first, and then multiply it by the derivative of the inner function.
For instance, if you have a composite function presented as \( s=(4+6t)^{3/2} \), and you need its derivative, you first identify the outer function \( f(u) = u^{3/2} \) and the inner function \( g(t) = 4 + 6t \).

• First, differentiate the outer function. If \( f(u) = u^{3/2} \), its derivative is \( f'(u) = \frac{3}{2}u^{1/2} \).
• Next, differentiate the inner function. For \( g(t) = 4 + 6t \), the derivative is \( g'(t) = 6 \).
• Finally, apply the chain rule: multiply \( f'(g(t)) \) by \( g'(t) \).

This chain rule approach helps us transition from position \( s \) to velocity \( v \) by understanding how changes in \( t \) affect \( s \).

Velocity represents how fast an object's position changes over time. In simpler terms, it's the speed of an object in a given direction. Calculating velocity involves differentiating the position function.
Given that \( s(t) = (4+6t)^{3/2} \), we apply the chain rule to find the velocity:

• The derivative of \( s \) with respect to \( t \) is \( v(t) = \frac{ds}{dt} = \frac{3}{2}(4+6t)^{1/2} \cdot 6 \).

To find the specific velocity at \( t=2 \), substitute 2 into the derivative:
\[ v(2) = \frac{3}{2}(4+6\cdot2)^{1/2} \cdot 6 \]
Calculating this gives us the velocity at that particular moment. Velocity is crucial for understanding motion as it provides insight into how rapidly a position changes, and in what direction.

Acceleration measures how the velocity of an object changes over time. It tells us how quickly something is speeding up or slowing down. Just like we found velocity by differentiating position, we find acceleration by differentiating velocity.
Our velocity function is \( v(t) = \frac{3}{2}(4+6t)^{1/2} \cdot 6 \). To compute acceleration, we differentiate this function with respect to \( t \). Again using the chain rule:

• The derivative of \( v(t) \) is \( a(t) = \frac{dv}{dt} = \frac{3}{4} \cdot 6 \cdot (4+6t)^{-1/2} \cdot 6 \).

To find the acceleration at \( t=2 \), substitute 2 into the acceleration formula:
\[ a(2) = \frac{3}{4} \cdot 6 \cdot (4+6\cdot2)^{-1/2} \cdot 6 \]
This calculation gives you the rate at which velocity changes at that instant. Understanding acceleration is key as it helps predict changes in motion, determining whether an object is speeding up or slowing down as it moves.