The only force acting on a 3.0 kg body as it moves along the positive x axis has an x component Fx = -9x N, where x is in meters. The velocity of the body at x = 2.3 m is 9.7 m/s. (a) What is the velocity of the body at x = 4.4 m? (b) At what positive value of x will the body have a velocity of 2.6 m/s?
To answer these questions, we will use Newton's second law of motion, which states that the force acting on an object is equal to the mass of the object multiplied by its acceleration. We can use this equation to find the acceleration, and then use it to calculate the velocity at different positions.
(a) To find the velocity of the body at x = 4.4 m, we first need to find the acceleration at that position. The force acting on the body is given by Fx = -9x N, where x is in meters. Since the force is in the x-direction, we can equate it to the mass of the body multiplied by its acceleration in the x-direction:
-9x = (3.0 kg) * ax
To find the acceleration (ax), we divide both sides of the equation by the mass:
ax = -9x / (3.0 kg)
ax = -3x
Now, we have the expression for the acceleration in terms of x. To find the velocity (vx) at x = 4.4 m, we need to integrate the acceleration with respect to x. Since the acceleration is -3x, we integrate it as follows:
∫ ax dx = ∫ (-3x) dx
Integrating this expression gives us the velocity function:
vx = -3 * (x^2) / 2 + C
Here, C is the constant of integration. To determine its value, we can use the given information that the velocity of the body at x = 2.3 m is 9.7 m/s. Substituting these values into the velocity function, we can solve for C:
9.7 = -3 * (2.3^2) / 2 + C
Simplifying this equation, we find that C = 23.53.
Now, we can use this value of C to find the velocity at x = 4.4 m:
vx = -3 * (4.4^2) / 2 + 23.53
Calculating this expression, we find that the velocity of the body at x = 4.4 m is 3.74 m/s.
(b) To find the positive value of x at which the body will have a velocity of 2.6 m/s, we set the velocity function equal to 2.6 m/s and solve for x:
2.6 = -3 * (x^2) / 2 + 23.53
Rearranging and solving this equation gives us:
-3 * (x^2) / 2 = 2.6 - 23.53
-3 * (x^2) / 2 = -20.93
x^2 = (2/3) * 20.93
x^2 = 13.95
x = √13.95
Taking the square root of both sides, we find that the positive value of x at which the body will have a velocity of 2.6 m/s is approximately 3.74 m.