A good model for the acceleration of a car trying to reach top speed in the least amount of time is ax = a0 ─ kvx, where a₀ is the initial acceleration and k is a constant. Find an expression for k in terms of a0 and the car's top speed vmax.

Ch 02: Kinematics in One Dimension

Knight Calc - Physics for Scientists and Engineers 5th Edition

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Knight Calc 5th Edition

Ch 02: Kinematics in One Dimension

Problem 81a

Chapter 2, Problem 81a

A good model for the acceleration of a car trying to reach top speed in the least amount of time is a_x = a₀ ─ kv_x, where a₀ is the initial acceleration and k is a constant. Find an expression for k in terms of a₀ and the car's top speed v_max.

Verified step by step guidance

Step 1: Start by analyzing the given equation for acceleration, a𝓍 = a₀ ─ kv𝓍. At the car's top speed vₘₐₓ, the acceleration a𝓍 becomes zero because the car can no longer increase its speed.

Step 2: Set a𝓍 = 0 in the equation, which gives 0 = a₀ ─ k * vₘₐₓ. This represents the condition at the car's maximum speed.

Step 3: Rearrange the equation to solve for k. Add k * vₘₐₓ to both sides, resulting in k * vₘₐₓ = a₀.

Step 4: Divide both sides of the equation by vₘₐₓ to isolate k. This gives k = a₀ / vₘₐₓ.

Step 5: Conclude that the expression for k in terms of a₀ and vₘₐₓ is k = a₀ / vₘₐₓ.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Acceleration

Acceleration is the rate of change of velocity of an object with respect to time. In this context, it describes how quickly the car can increase its speed. The initial acceleration (a₀) represents the car's acceleration at the start, while the model suggests that this acceleration decreases as the car's speed increases, influenced by the constant k.

Top Speed

Top speed, denoted as vₘₐₓ, is the maximum velocity that a car can achieve under ideal conditions. It is a critical parameter in the model, as it defines the limit of acceleration. Understanding the relationship between initial acceleration and top speed is essential for deriving the constant k, which quantifies how acceleration diminishes as the car approaches its maximum speed.

Exponential Decay

The equation a𝓍 = a₀ - kv𝓍 represents a form of exponential decay, where the acceleration decreases as the velocity increases. The constant k determines the rate of this decay, indicating how quickly the car's acceleration diminishes as it approaches its top speed. This concept is vital for understanding the dynamics of the car's motion and how forces interact as speed increases.

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A good model for the acceleration of a car trying to reach top speed in the least amount of time is a_𝓍 = a_₀ ─ kv_𝓍, where a_₀ is the initial acceleration and k is a constant. Find an expression for the car's velocity as a function of time.

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