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In accelerating from a standing start to a speed of \(300 \mathrm{km} / \mathrm{h}(186 \mathrm{mi} / \mathrm{h}-\text { not its top speed! }),\) the 1900 -kg Bugatti Veyron sports car exerts an average force on the road of \(9.4 \mathrm{kN}\). How long does the car take to reach \(300 \mathrm{km} / \mathrm{h} ?\)

Short Answer

Expert verified
The car takes approximately 16.84 seconds to reach 300 km/h.

Step by step solution

01

- Convert the Speed to Meters Per Second

First, convert the speed from km/h to m/s using the formula: \[ 1 \text{ km/h} = \frac{1}{3.6} \text{ m/s} \] Thus, \[ 300 \text{ km/h} = 300 \times \frac{1}{3.6} \text{ m/s} = 83.33 \text{ m/s} \]
02

- Calculate the Acceleration

Use Newton's second law to find the acceleration: \[ F = ma \] Given a force ( \( F = 9.4 \text{ kN} = 9400 \text{ N} \) ) and mass ( \( m = 1900 \text{ kg} \) ), we find the acceleration ( \( a \) ) as: \[ a = \frac{F}{m} = \frac{9400 \text{ N}}{1900 \text{ kg}} = 4.95 \text{ m/s}^2 \]
03

- Calculate the Time

Use the kinematic equation to find the time: \[ v = u + at \] Since the car starts from rest ( \( u = 0 \) ), the equation simplifies to: \[ t = \frac{v}{a} = \frac{83.33 \text{ m/s}}{4.95 \text{ m/s}^2} = 16.84 \text{ s} \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Kinematic Equations
Kinematic equations describe the motion of objects without considering the forces that cause the motion. These equations are essential in solving problems involving velocity, acceleration, displacement, and time.

In our problem, we use the equation for velocity: \[ v = u + at \]

Here, \(( v \) represents the final velocity, \(( u \) the initial velocity, \(( a \) the acceleration, and \(( t \) the time taken. Since the car starts from rest, the initial velocity \(( u \) is zero, simplifying our equation to: \[ t = \frac{v}{a} \]

This formula helps us calculate the time taken for the car to reach a speed of 300 km/h. Understanding this concept is vital for solving similar problems in physics. It's remarkable how these simple equations can predict the behavior of objects in motion!
Newton's Second Law
Newton's Second Law of Motion states that the force acting on an object is equal to the mass of the object multiplied by its acceleration \[ ( F = ma ) \].

In our scenario, the Bugatti Veyron exerts a force of 9400 N (kilonewtons converted to newtons) on the road. Using the car's mass (1900 kg), we calculate the acceleration by rearranging the formula: \[ a = \frac{F}{m} \]

Plugging in our values: \[ a = \frac{9400 \text{ N}}{1900 \text{ kg}} = 4.95 \text{ m/s}^2 \]

This concept is fundamental to understanding how objects respond to various forces. It tells us that more force results in more acceleration, and more mass leads to less acceleration for the same force. Newton's Second Law is a cornerstone of classical mechanics and explains how the dynamics of motion operate.
Unit Conversion
Unit conversion is an essential skill in physics to ensure all measurements are in compatible units.

In our problem, we convert the speed from km/h to m/s using the conversion factor: \[ 1 \text{ km/h} = \frac{1}{3.6} \text{ m/s} \]

This conversion is crucial to solving the problem accurately. If we didn't convert units, our calculations for time and acceleration would be incorrect. Here's the conversion step for our speed: \[ 300 \text{ km/h} = 300 \times \frac{1}{3.6} \text{ m/s} = 83.33 \text{ m/s} \]

Remembering to convert units ensures that you avoid common mistakes and helps in achieving the correct outcome. Whether dealing with length, mass, or speed, unit conversion aligns different measurements, making calculations viable. Practice makes perfect, and soon these conversions will become second nature!

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Most popular questions from this chapter

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