Question

A particle moving ina straight line along the x axis has velocity related to x as v=16rootx what is the acceleration
a) 16m
b)128m
c) 256m
d)64m
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Answer 1

ACCELERATION IS 128 m/s² (Option b)

Given:

• Velocity function with respect to Displacement is v = 16√x

To find:

Value of acceleration?

Calculation:

We will use Calculus to find out the acceleration value:

$a = \dfrac{dv}{dt}$

• Applying CHAIN RULE OF DIFFERENTIATION:

$\implies a = \dfrac{dv}{dx} \times \dfrac{dx}{dt}$

$\implies a = \dfrac{dv}{dx} \times v$

• Putting proper functions:

$\implies a = \dfrac{d(16 \sqrt{x} )}{dx} \times 16 \sqrt{x}$

$\implies a = \dfrac{16 {x}^{ - \frac{1}{2} } }{2} \times 16 \sqrt{x}$

$\implies a = \dfrac{8 }{ \sqrt{x} } \times 16 \sqrt{x}$

$\implies a = 128 \: m {s}^{ - 2}$

Acceleration is 128 m/s².

Answer 2

Concept:

• Differentiation
• One-dimensional motion
• Kinematics equation
• Differentiating displacement and velocity to get velocity and acceleration respectively

Given:

• velocity v = 16 √x m/s
• displacement = x m

Find:

• the acceleration

Solution:

Acceleration is related to velocity in the following way

a = dv/dt

where dv is an extremely small change in velocity and dt is an extremely short period of time

We have to use the chain rule differentiation

a = (dv/dx) (dx/dt)

v = 16 √x

dv/dx = d (16 √x)/dx

dv/dx = 16 d(√x/dx)

dv/dx = 16(1/2√x)

dv/dx = 8/√x

velocity v = dx/dt

a = (dv/dx) (dx/dt) can be rewritten as

a = (dv/dx) v

a = 8/√x (16 √x)

a = 128 m/s^2

The acceleration is 128 m/s^2.

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