The only force acting on a 4.7 kg body as it moves along the positive x axis has an x component Fx = -6x N, where x is in meters. The velocity of the body at x = 2.4 m is 11 m/s. (a) What is the velocity of the body at x = 4.9 m? (b) At what positive value of x will the body have a velocity of 3.5 m/s?
To find the velocity of the body at different positions along the x-axis, we need to use Newton's second law of motion, which states that the net force acting on an object is equal to the product of its mass and acceleration.
In this case, the net force is given by Fx = -6x N, where x is the position along the x-axis. The mass of the body is 4.7 kg.
(a) To find the velocity of the body at x = 4.9 m, we can use the concept of work and energy. The work done on an object is equal to the change in its kinetic energy. Mathematically, we can write:
Work = ΔKE
The work done by the net force is given by the equation:
Work = ∫ F dx
where ∫ represents integration and F represents the net force.
Integrating the given force with respect to x, we get:
Work = ∫ -6x dx
Integrating -6x with respect to x, we get:
Work = -3x^2 + C
where C is the constant of integration.
Since the work done is equal to the change in kinetic energy, we can write:
-3x^2 + C = ΔKE
When x = 2.4 m, the velocity of the body is 11 m/s. Thus, the kinetic energy at x = 2.4 m is given by:
KE1 = (1/2)mv^2
Substituting the given values, we get:
KE1 = (1/2)(4.7 kg)(11 m/s)^2
Now let's find the value of C. At x = 2.4 m, the work done is equal to the change in kinetic energy:
-3(2.4 m)^2 + C = (1/2)(4.7 kg)(11 m/s)^2
Simplifying the equation, we find:
C = (1/2)(4.7 kg)(11 m/s)^2 + 3(2.4 m)^2
Now, we can find the kinetic energy (KE2) at x = 4.9 m by using the equation:
KE2 = -3x^2 + C
Substituting x = 4.9 m and the value of C, we can calculate the kinetic energy at x = 4.9 m.
Finally, the velocity of the body at x = 4.9 m can be found by using the equation:
v = sqrt(2KE/m)
where v is velocity, KE is kinetic energy, and m is mass.
(b) To find the positive value of x at which the body will have a velocity of 3.5 m/s, we can use the same equation for kinetic energy as in part (a):
KE = (1/2)mv^2
Solving for x, we can find the positive value of x at which the body has a velocity of 3.5 m/s.