Suppose that the velocity of a parachutist is v(t)=65(1-e^-0.16 t) meters per second. The graph

step 1 In this question we recall that the acceleration function AT will equal to the V prime under T.

Suppose that the velocity of a parachutist is...

Suppose that the velocity of a parachutist is
$$v(t)=65\left(1-e^{-0.16 t}\right)$$
meters per second. The graph of $v(t)$ is similar to that in Fig. 2 Calculate the parachutist's velocity and acceleration when $t=9$ seconds.

Key Concepts

Kinematics

Kinematics is the branch of mechanics that describes the motion of objects without considering the causes of motion. Key aspects include quantities such as displacement, velocity, and acceleration. In problems like this one, one often uses a given velocity function and differentiates it to find the acceleration, which describes how the velocity changes over time.

Exponential Functions

Exponential functions, such as those involving terms like e^(?at), are common in modeling processes that exhibit decay or growth. In this context, an exponential function is used to describe how the velocity approaches a limiting value over time. Understanding the properties of the exponential function is crucial for solving problems involving exponential decay or asymptotic behavior.

Differentiation (Chain Rule)

Differentiation is a core concept in calculus used to determine the rate at which one quantity changes with respect to another. When the function involves a composite expression such as e^(?0.16t), the chain rule is applied. The chain rule allows one to differentiate composite functions by taking the derivative of the outer function and then multiplying it by the derivative of the inner function.

Function Evaluation

Function evaluation involves substituting a specific value for the variable into a function and computing the output. This is used here to determine the exact values of both velocity and acceleration at a given time. It connects the abstract functional description of motion to concrete numerical results that describe the real-world behavior at specific instants.

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