Ch 02: Kinematics in One Dimension

Knight Calc - Physics for Scientists and Engineers 5th Edition

Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Not the one you use?Change textbook

Knight Calc 5th Edition

Ch 02: Kinematics in One Dimension

Problem 81b

Chapter 2, Problem 81b

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.

Verified step by step guidance

Start with the given differential equation for acceleration:

aₓ = a₀ - kvₓ

. Recall that acceleration is the derivative of velocity with respect to time, so rewrite it as

dvₓ/dt = a₀ - kvₓ

Rearrange the equation to isolate terms involving

vₓ

and

t

dvₓ/(a₀ - kvₓ) = dt

. This sets up the equation for integration.

Integrate both sides. On the left-hand side, integrate with respect to

vₓ

, and on the right-hand side, integrate with respect to

t

. The integral of the left-hand side is

-ln|a₀ - kvₓ|/k

, and the integral of the right-hand side is

t + C

, where

C

is the constant of integration.

Solve for

vₓ

. Exponentiate both sides to remove the logarithm:

a₀ - kvₓ = e^{-k(t + C)}

. Rearrange to isolate

vₓ

vₓ = (a₀/k) - (1/k)e^{-kt}e^{-kC}

Simplify the expression further by combining constants. Let

e^{-kC}

be represented as a new constant

C'

. The final expression for velocity as a function of time is

vₓ(t) = (a₀/k) - (C'/k)e^{-kt}

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Key Concepts

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

Differential Equations

Differential equations are mathematical equations that relate a function to its derivatives. In this context, the equation a𝓍 = a₀ - kv𝓍 describes how acceleration changes with velocity, which can be expressed as a first-order differential equation. Solving this equation will allow us to find the velocity of the car as a function of time.

Acceleration and Velocity Relationship

Acceleration is defined as the rate of change of velocity with respect to time. In the given model, the acceleration of the car decreases as its velocity increases, which is represented by the term -kv𝓍. Understanding this relationship is crucial for deriving the velocity function, as it shows how the car's speed approaches its maximum as time progresses.

Initial Conditions

Initial conditions are the values of a function and its derivatives at a specific point, often used to solve differential equations. In this problem, the initial acceleration a₀ serves as a starting point for the car's motion. Knowing the initial conditions allows us to integrate the differential equation correctly and find the specific solution for the car's velocity over time.

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Example

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Textbook Question

A rocket is launched straight up with constant acceleration. Four seconds after liftoff, a bolt falls off the side of the rocket. The bolt hits the ground 6.0 s later. What was the rocket's acceleration?

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Textbook Question

A rubber ball is shot straight up from the ground with speed v₀. Simultaneously, a second rubber ball at height h directly above the first ball is dropped from rest. At what height above the ground do the balls collide? Your answer will be an algebraic expression in terms of h, v₀, and g.

Textbook Question

A sprinter can accelerate with constant acceleration for 4.0 s before reaching top speed. He can run the 100 meter dash in 10.0 s. What is his speed as he crosses the finish line?

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Textbook Question

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.

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Textbook Question

A rocket in deep space has an empty mass of 150 kg and exhausts the hot gases of burned fuel at 2500m/s . It is loaded with 600 kg of fuel, which it burns in 30 s. What is the rocket's speed 10 s, 20 s, and 30 s after launch?

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Textbook Question

Careful measurements have been made of Olympic sprinters in the 100 meter dash. A quite realistic model is that the sprinter's velocity is given by v_𝓍 = a ( 1 - e⁻ᵇᵗ ) where t is in s, v_𝓍 is in m/s, and the constants a and b are characteristic of the sprinter. Sprinter Carl Lewis's run at the 1987 World Championships is modeled with a = 11.81 m/s and b = 0.6887 s⁻¹. Find an expression for the distance traveled at time t.

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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 a𝓍 = a₀ ─ kv𝓍, where a₀ is the initial acceleration and k is a constant. Find an expression for the car's velocity as a function of time.

Verified video answer for a similar problem:

Key Concepts

Differential Equations

Acceleration and Velocity Relationship

Initial Conditions

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.