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Chapter 2: Problem 225

A particle moves in straight line. Its position is given by $\mathrm{x}=2+5 \mathrm{t}-3 \mathrm{t}^{2}$. Find the ratio of initial velocity and initial acceleration. \((\mathrm{A})+(5 / 6)\) (B) \(-(5 / 6)\) (C) \((6 / 5)\) (D) \(-(6 / 5)\)

Short Answer

Expert verified
The ratio of initial velocity and initial acceleration is \(-\dfrac{5}{6}\).

Step by step solution

01

Find the velocity function

To find the velocity function, take the first derivative of the position function with respect to time: \(v(t) = \dfrac{d}{dt}(2 + 5t - 3t^2)\) \(v(t) = 5 - 6t\)
02

Find the initial velocity

To find the initial velocity, substitute t=0 into the velocity function: \(v(0) = 5 - 6(0) = 5\)
03

Find the acceleration function

To find the acceleration function, take the second derivative of the position function with respect to time: \(a(t) = \dfrac{d^2}{dt^2}(2 + 5t - 3t^2)\) \(a(t) = -6\) Since acceleration is constant, the initial acceleration is also -6.
04

Find the ratio of initial velocity and initial acceleration

To find the ratio of initial velocity and initial acceleration, divide the initial velocity by the initial acceleration: Ratio = \(\dfrac{v(0)}{a(0)} = \dfrac{5}{-6}\) So the ratio is \(-\dfrac{5}{6}\). The correct answer is (B) \(-\dfrac{5}{6}\).

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Most popular questions from this chapter

Comprehensions type questions. A particle is moving in a circle of radius \(\mathrm{R}\) with constant speed. The time period of the particle is T Now after time \(\mathrm{t}=(\mathrm{T} / 6)\) Average speed of the particle is (A) \((\pi \mathrm{R} / 6 \mathrm{~T})\) (B) \([(2 \pi R) / 3 \mathrm{~T}]\) (C) \([(2 \pi R) / T]\) (D) \((\mathrm{R} / \mathrm{T})\)

Motion of a particle is described by \(\mathrm{x}=(\mathrm{t}-2)^{2}\) Find its velocity when it passes through origin. (A) \(0 \mathrm{~m} / \mathrm{s}\) (B) \(2 \mathrm{~ms}^{-1}\) (C) \(4 \mathrm{~ms}^{-1}\) (D) \(8 \mathrm{~ms}^{-1}\)

The motion of a particle along a straight line is described by the function \(\mathrm{x}=(3 \mathrm{t}-2)^{2}\). Calculate the acceleration after $10 \mathrm{~s}$. (A) \(9 \mathrm{~ms}^{-2}\) (B) \(18 \mathrm{~ms}^{-2}\) (C) \(36 \mathrm{~ms}^{-2}\) (D) \(6 \mathrm{~ms}^{-2}\)

A particle is thrown in upward direction with initial velocity of $60 \mathrm{~m} / \mathrm{s}$. Find average speed and average velocity after 10 seconds. \(\left[\mathrm{g}=10 \mathrm{~ms}^{-2}\right]\) (A) \(26 \mathrm{~ms}^{-1}, 16 \mathrm{~ms}^{-1}\) (B) \(26 \mathrm{~ms}^{-1}, 10 \mathrm{~ms}^{-1}\) (C) \(20 \mathrm{~ms}^{-1}, 10 \mathrm{~ms}^{-1}\) (D) \(15 \mathrm{~ms}^{-1}, 25 \mathrm{~ms}^{-1}\)

$\quad\left(\mathrm{A}^{\rightarrow}+\mathrm{B}^{\rightarrow}\right) \cdot\left(\mathrm{A}^{\rightarrow} \times \mathrm{B}^{\rightarrow}\right)=\ldots$ (A) 0 (B) \(A^{2}+B^{2}\) (C) \(\sqrt{\left(A^{2}+B^{2}\right)}\) (D) \(\mathrm{A}^{2} \mathrm{~B}^{2}\)
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