Textbook Question
The accompanying figure shows the velocity v = ds/dt = f(t) (m/sec) of a body moving along a coordinate line.
a. When does the body reverse direction?
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4. Applications of Derivatives
Motion Analysis
5:42 minutesProblem 7.3.97a
Textbook Question
Terminal velocity Refer to Exercises 95 and 96.
a. Compute a jumper’s terminal velocity, which is defined as lim t → ∞ v(t) = lim t → ∞ √(mg/k) tanh (√(kg/m) t).
Verified step by step guidance1
Identify the given velocity function: \(v(t) = \sqrt{\frac{mg}{k}} \tanh \left( \sqrt{\frac{kg}{m}} t \right)\), where \(m\) is mass, \(g\) is acceleration due to gravity, and \(k\) is a constant related to air resistance.
Recall the definition of terminal velocity as the limit of \(v(t)\) as time \(t\) approaches infinity: \(\lim_{t \to \infty} v(t)\).
Focus on the behavior of the hyperbolic tangent function \(\tanh(x)\) as \(x \to \infty\). Remember that \(\tanh(x)\) approaches 1 when \(x\) becomes very large.
Apply this limit to the velocity function: replace \(\tanh \left( \sqrt{\frac{kg}{m}} t \right)\) with 1 as \(t \to \infty\).
Conclude that the terminal velocity is \(\lim_{t \to \infty} v(t) = \sqrt{\frac{mg}{k}} \times 1 = \sqrt{\frac{mg}{k}}\).
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5mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Limit of a Function as t Approaches Infinity
The limit of a function as t approaches infinity describes the behavior of the function as the input grows without bound. In this problem, evaluating lim t → ∞ v(t) helps find the steady-state velocity, known as terminal velocity, where acceleration ceases and velocity stabilizes.
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Hyperbolic Tangent Function (tanh)
The hyperbolic tangent function, tanh(x), is a smooth, continuous function that approaches 1 as x approaches infinity and -1 as x approaches negative infinity. Understanding tanh's limiting behavior is crucial to simplifying the velocity expression for large time values.
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Terminal Velocity in Physics
Terminal velocity is the constant speed an object reaches when the force of gravity is balanced by the drag force, resulting in zero net acceleration. Mathematically, it is the limit of the velocity function as time goes to infinity, representing the maximum velocity attainable during free fall.
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