An object with a mass of $2 kg$ is traveling at $14 m/s$ . If the object is accelerated by a force of $f(x) = x+xe^x$ over $x in [0, 9]$ , where x is in meters, what is the impulse at $x = 8$ ?

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Answer 1 · 2016-03-24T22:12:16

1

The definition of impulse is

$I=F*Deltat =$

and using the 2nd law of Newton $F=m*a$ it is equivalent to the variation of momentum,

$I=Deltap = m*Deltav = m*(v_f -v_0) ; m=2kg ; v_0=14m/s$

so, we need to calculate the speed at the position $x=8; v_f$

By the 2nd law of Newton $F=m*a => F(x) = m* (dv)/dt$

$F(x) = m*(dv(x))/dx * (dx)/dt = m *(dv(x))/dx *v(x)$

$int_0^8F(x)dx = m*(v_f^2 -v_0^2)$

$v_f = sqrt (v_0^2+1/m*int_0^8F(x)dx)$

thus, the impulse is

$I=m*(sqrt (v_0^2+1/m*int_0^8F(x)dx) -v_0) = 262.5* kg* m/s$

An object with a mass of 2 kg2 kg is traveling at 14 m/s14 m/s. If the object is accelerated by a force of f(x) = x+xe^x f(x) = x+xe^x over x in [0, 9]x in [0, 9], where x is in meters, what is the impulse at x = 8x = 8?