The only force acting on a 3.1 kg body as it moves along the positive x axis has an x component fx = - 8x n, where x is in meters. the velocity of the body at x = 2.0 m is 11 m/s. (a) what is the velocity of the body at x = 4.5 m? (b) at what positive value of x will the body have a velocity of 5.8 m/s?

(a) - 9.97 m/s

(b) x = 2.83

This is a simple problem in integral calculus. You've been given part of the 2nd derivative (acceleration), but not quite. You've been given the force instead. So let's setup a function for acceleration.

f'' (x) = - 8x N / 3.1 kg = - 8x kg*m/s^2 / 3.1 kg = - 2.580645161x m/s^2

So the acceleration of the body is now expressed as

f'' (x) = - 2.580645161x m/s^2

Let's calculate the anti-derivative from that.

f'' (x) = - 2.580645161x m/s^2

f' (x) = - 1.290322581x^2 + C m/s

Now let's use the known velocity value at x = 2.0 to calculate C

f' (x) = - 1.290322581x^2 + C 1

1 = - 1.290322581*2^2 + C

11 = - 1.290322581*4 + C

11 = - 5.161290323 + C

16.161290323 = C

So the velocity function is

f' (x) = - 1.290322581x^2 + 16.161290323

(a) The velocity at x = 4.5

f' (x) = - 1.290322581x^2 + 16.161290323

f' (4.5) = - 1.290322581*4.5^2 + 16.161290323

f' (4.5) = - 1.290322581*20.25 + 16.161290323

f' (4.5) = - 26.12903227 + 16.161290323

f' (4.5) = - 9.967741942

So the velocity is - 9.97 m/s

(b) we want a velocity of 5.8 m/s

5.8 = - 1.290322581x^2 + 16.161290323

0 = - 1.290322581x^2 + 10.36129032

1.290322581x^2 = 10.36129032

x^2 = 8.029999998

x = 2.833725463