An object with a mass of $6 k g$ $6 kg$ is traveling at $2 \frac{m}{s}$ $2 m/s$ . If the object is accelerated by a force of $f (x) = e^{2 x}$ $f(x) = e^(2x)$ over $x \in [0, 9]$ $x in [0, 9]$ , where x is in meters, what is the impulse at $x = 6$ $x = 6$ ?

Question

An object with a mass of $6 k g$ $6 kg$ is traveling at $2 \frac{m}{s}$ $2 m/s$ . If the object is accelerated by a force of $f (x) = e^{2 x}$ $f(x) = e^(2x)$ over $x \in [0, 9]$ $x in [0, 9]$ , where x is in meters, what is the impulse at $x = 6$ $x = 6$ ?

1 Answer

Impact of this question

1514 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2016-02-07T21:10:47

let's write the impulse equation:
$F (x) = m \frac{d v}{d t}$ $F(x) = m (dv)/dt$ re-write in terms of x
$F (x) = m \frac{d^{2} x}{{d t}^{2}}$ $F(x) = m (d^2x)/dt^2$
we have a differential equation of the form:
$\frac{d^{2} x}{{d t}^{2}} = \frac{1}{m} F (x, \frac{d x}{d t})$ $(d^2x)/dt^2 = 1/mF(x, dx/dt)$
Let $u = \frac{d x}{d t};$ $u = dx/dt;$
using chain rule $\frac{d u}{d t} = \frac{d u}{d x} \frac{d x}{d t} = u \frac{d u}{d x}$ $(du)/dt = (du)/dx (dx)/dt = u (du)/dx$
$u \frac{d u}{d x} = \frac{1}{m} F (x, u) = \frac{1}{m} f (x)$ $u (du)/dx = 1/mF(x, u) = 1/m f(x)$ the force depends on x only

$u \frac{d u}{d x} = \frac{1}{m} f (x) = e^{2 x}$ $u (du)/dx = 1/m f(x) = e^(2x)$ rearrange and solve by integrating
$\int u d u = \frac{1}{m} \int e^{2 x} d x$ $int udu = 1/m inte^(2x) dx$
$\frac{1}{2} u^{2} = \frac{1}{m} e^{2 x} + C$ $1/2u^2 = 1/m e^(2x) + C$
since the object started with a velocity of 2m/s
$C = 4;$ $C = 4;$ and initial velocity, $v_{o} = \sqrt{C} = 2$ $v_o = sqrt(C) = 2$
$v = \sqrt{\frac{1}{3} e^{2 x}} + 2$ $v = sqrt(1/3 e^(2x) ) + 2$

Evaluate velocity at $x = 6; v_{x = 6} = \sqrt{\frac{1}{3} e^{2 \cdot 6}} + 2 \approx 235$ $x = 6; v_(x=6) = sqrt(1/3e^(2*6)) +2 ~~235$
Now Impulse, $I = m (v_{2} - v_{1}) \approx 6 (233) = 1398 K g - \frac{m}{s}$ $I = m (v_2-v_1) ~~ 6(233) = 1398 Kg-m/s$

Science

Math

Humanities

... and beyond

An object with a mass of $6 k g$ $6 kg$ is traveling at $2 \frac{m}{s}$ $2 m/s$ . If the object is accelerated by a force of $f (x) = e^{2 x}$ $f(x) = e^(2x)$ over $x \in [0, 9]$ $x in [0, 9]$ , where x is in meters, what is the impulse at $x = 6$ $x = 6$ ?

1 Answer

Related questions

Impact of this question

Science

Math

Humanities

... and beyond

An object with a mass of 6kg6 kg is traveling at 2ms2 m/s. If the object is accelerated by a force of f(x)=e2xf(x) = e^(2x) over x∈[0,9]x in [0, 9], where x is in meters, what is the impulse at x=6x = 6?

1 Answer

Related questions

Impact of this question

An object with a mass of $6 k g$ $6 kg$ is traveling at $2 \frac{m}{s}$ $2 m/s$ . If the object is accelerated by a force of $f (x) = e^{2 x}$ $f(x) = e^(2x)$ over $x \in [0, 9]$ $x in [0, 9]$ , where x is in meters, what is the impulse at $x = 6$ $x = 6$ ?