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Chapter 3: Problem 43

The rate of change of velocity is acceleration.

Short Answer

Expert verified
Answer: The acceleration function is \(a(t) = 6t + 2\).

Step by step solution

01

Understanding Velocity

Velocity is a measure of an object's speed in a specific direction. It is a vector quantity, which means it has both magnitude and direction. In physics, velocity is typically represented by the symbol \(v\) and is measured in units of meters per second (m/s).
02

Understanding Acceleration

Acceleration is the rate of change of velocity with respect to time. It is a measure of how quickly an object's velocity is changing. Acceleration is also a vector quantity, with the symbol \(a\) and its units are meters per second squared (m/s\(^2\)).
03

Relationship between Velocity and Acceleration

The relationship between velocity and acceleration is expressed through the derivative. The acceleration of an object is the derivative of its velocity with respect to time, which can be mathematically represented as: \(a=\frac{dv}{dt}\)
04

Example: Calculating Acceleration from a given Velocity Function

Let's consider an example where the velocity function \(v(t)\) of a moving object is given by: \(v(t)= 3t^2+2t-4\). To find the acceleration function, \(a(t)\), we need to take the derivative of the given velocity function with respect to time, \(t\).
05

Taking the Derivative of the Velocity Function

To find the acceleration function, we differentiate the given velocity function with respect to time: \(a(t)=\frac{d}{dt}(3t^2+2t-4)\)
06

Solving for Acceleration

Now, we can apply the rules of differentiation to find the acceleration function: \(a(t)=\frac{d}{dt}(3t^2)+\frac{d}{dt}(2t)-\frac{d}{dt}(4)\) Applying the power rule and the constant rule of differentiation, we get: \(a(t) = 6t + 2\)
07

Final Result

The acceleration function for the given velocity function \(v(t)=3t^2+2t-4\) is \(a(t) = 6t + 2\).

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Most popular questions from this chapter

Fill in the Blanks. \(\frac{\mathrm{T}}{2}\) The time taken to perform one to-and-fro motion (or) from one extreme position to other extreme position and back is called time period (T).

Fill in the Blanks. vibratory The molecules in solid undergo vibratory motion.

From the given figure, by pythogorous theorem \(\mathrm{PR}^{2}=\mathrm{PQ}^{2}+\mathrm{R} \mathrm{Q}^{2}\) \(\mathrm{PR}^{2}=4^{2}+3^{2}\) \(\mathrm{PR}^{2}=16+9\) \(\mathrm{PR}=\sqrt{25}\) \(\mathrm{PR}=5 \mathrm{~m}\) Average speed \(=\frac{\text { total distance }}{\text { total time }}=\frac{4 \mathrm{~m}+3 \mathrm{~m}+5 \mathrm{~m}}{20 \mathrm{~s}+10 \mathrm{~s}+30 \mathrm{~s}}=\frac{12 \mathrm{~m}}{60 \mathrm{~s}}=0.2 \mathrm{~m} \mathrm{~s}^{-1}\)

The speed and average speed of the vehicle can be equal if the vehicle moves with uniform speed or constant speed.

\(\mathrm{A} \rightarrow \mathrm{b} \quad\) The piston of a motorcar engine moving at uniform speed is said to be in periodic motion. \(\mathrm{B} \rightarrow \mathrm{e}, \mathrm{b} \quad\) The objects executing vibratory motion undergo change in shape or size. The piston of a motor car engine executes vibratory motion. \(\mathrm{C} \rightarrow \mathrm{g} \quad\) Body at rest will have zero speed as well as zero velocity. \(\mathrm{D} \rightarrow\) a Maximum displacement of a body from its mean position is called amplitude. \(\mathrm{E} \rightarrow \mathrm{c} \quad\) A body moving with variable speed is said to be in non-uniform motion. \(\mathrm{F} \rightarrow \mathrm{d} \quad 1 \mathrm{~ms}^{-1}=\frac{1 \mathrm{~m}}{1 \mathrm{~s}}=\frac{\frac{1}{100} \mathrm{~km}}{\frac{1}{3600} \mathrm{~h}}=\frac{18}{5} \mathrm{~km} \mathrm{~h}^{-1}\) \(\mathrm{G} \rightarrow \mathrm{f} \quad\) Average velocity \(=\frac{\text { Total displacement }}{\text { Total time }}\)
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