(C) Given the position function: $x(t) = 4t^3 - 3t^2 + 2$. Velocity $v(t)$ is the first derivative of position with respect to time: $v(t) = \frac{dx}

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$42 \, ms^{-2}$ and $36 \, ms^{-1}$

(C) Given the position function: $x(t) = 4t^3 - 3t^2 + 2$. Velocity $v(t)$ is the first derivative of position with respect to time: $v(t) = \frac{dx}{dt} = \frac{d}{dt}(4t^3 - 3t^2 + 2) = 12t^2 - 6t$. Acceleration $a(t)$ is the derivative of velocity with respect to time: $a(t) = \frac{dv}{dt} = \frac{d}{dt}(12t^2 - 6t) = 24t - 6$. At $t = 2 \, s$: Velocity $v(2) = 12(2)^2 - 6(2) = 12(4) - 12 = 48 - 12 = 36 \, ms^{-1}$. Acceleration $a(2) = 24(2) - 6 = 48 - 6 = 42 \, ms^{-2}$. Therefore,the acceleration is $42 \, ms^{-2}$ and the velocity is $36 \, ms^{-1}$.

Class 11 Physics Motion in a Straight Line Instantaneous Velocity and Speed and Velocity-time Graph

Medium

The position $x$ of a particle at any time $t$ is given by $x(t) = 4t^3 - 3t^2 + 2$. The acceleration and velocity of the particle at $t = 2 \, s$ are respectively:

AIIMS 2009 All AIIMS

A
$16 \, ms^{-2}$ and $22 \, ms^{-1}$
B
$48 \, ms^{-2}$ and $36 \, ms^{-1}$
C
$42 \, ms^{-2}$ and $36 \, ms^{-1}$
D
$12 \, ms^{-2}$ and $25 \, ms^{-1}$

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Motion in a Straight Line

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Instantaneous Velocity and Speed and Velocity-time Graph

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The acceleration versus velocity graph of a particle moving in a straight line,starting from rest,is as shown in the figure. The corresponding velocity-time graph would be:

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The position $x$ of a particle at any time $t$ is given by $x(t) = 4t^3 - 3t^2 + 2$. The acceleration and velocity of the particle at $t = 2 \, s$ are respectively:

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We have carefully distinguished between average speed and magnitude of average velocity. No such distinction is necessary when we consider instantaneous speed and magnitude of velocity. The instantaneous speed is always equal to the magnitude of instantaneous velocity. Why?

The displacement-time graphs of two moving particles make angles of $30^{\circ}$ and $45^{\circ}$ with the time-axis as shown in the figure. The ratio of their respective velocities is:

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