A Roller Coaster's Sudden Stop?

📌 CONCEPT: The instantaneous rate of change of the velocity of the roller coaster at the instant it comes to a stop is its instantaneous acceleration, which is the derivative of its velocity function at that particular time, and the speed at which it comes to a stop is the value of the velocity function when time equals the time taken for it to stop.

📐 RULE / FORMULA: To find the instantaneous acceleration, we need to find the derivative of the given velocity function v(t) = 20e^(-t/2) and evaluate it at the given time t = 2 seconds.

💡 WORKED EXAMPLE: Let's find the instantaneous acceleration of the roller coaster at t = 2 seconds. First, we find the derivative of v(t) = 20e^(-t/2) using the chain rule, which is v'(t) = -10e^(-t/2). Then, we evaluate v'(2) = -10e^(-2/2) = -10e^(-1). The instantaneous acceleration is -10e^(-1) m/s^2, and the speed at that instant is v(2) = 20e^(-2/2) = 20e^(-1) m/s.

⚠️ COMMON MISTAKE: Students may forget to include the negative sign when evaluating the derivative of the velocity function or may not consider the units of the instantaneous acceleration, which in this case is m/s^2.