Question

The acceleration of a body moving along x-axis is given by a=4x^3 ,where a is in m/s^2 and x is in metre.If at x=0 the velocity of body is 2m/s,then find its velocity at x=4 m.

Answer 1

The acceleration of a body moving along x-axis is given by $a = 4x^3$, where $a$ is in m/s^2 and $x$ is in metre. If at $x = 0$ the velocity of body is $2\mathrm{m/s}$, then find its velocity at $x = 4\mathrm{m}$.

$$a = 4 x ^ {3}$$

By definition:

$$a = \frac {d v}{d t} = \frac {d v}{d x} \frac {d x}{d t} = \frac {d v}{d x} v$$

Separate the variables:

$$4 x ^ {3} d x = v d v$$

Integrate:

$$x ^ {4} + C = \frac {v ^ {2}}{2}$$ $$v (x) = \pm \sqrt {2} \sqrt {x ^ {4} + C}$$

Initial condition: $v(0) = 2$

$$v (0) = \sqrt {2} \sqrt {0 ^ {4} + C} = 2$$ $$C = 2$$

Therefore,

$$v (x) = \sqrt {2} \sqrt {x ^ {4} + 2}$$

Finally,

$$v (4) = \sqrt {2} \sqrt {4 ^ {4} + 2} = 2 \sqrt {1 2 9} \frac {m}{s}$$

Answer: $v(4) = 2\sqrt{129}\frac{m}{s}$