Question

A particle of mass 5.2 kg is constrained to move along the x-axis under a single force

F(x) = −cx3,
where
c = 9.6 N/m3.
The particle's speed at A, where
xA = 1.0 m,
is 7.8 m/s. What is its speed (in m/s) at B, where
xB = −2.0 m?

Answer 1

To determine the speed of the particle at point B, we apply the conservation of mechanical energy, accounting for the work done by the force while moving from point A to B. By calculating the difference in kinetic energy and solving for speed at point B, we can find the desired value.

Step-by-step explanation:

Considering a particle of mass 5.2 kg is moving along the x-axis with a force F(x) = −cx^3, where c = 9.6 N/m3. To find the particle's speed at point B (xB = −2.0 m), we use the work-energy principle. Since there are no dissipative forces, the mechanical energy is conserved.

The work done by the force as the particle moves from A to B will change the particle's kinetic energy.

At A (xA = 1.0 m), the kinetic energy is (1/2)mvA2 and at B (xB = −2.0 m), the kinetic energy is (1/2)mvB2. We calculate the work done W by integrating F(x) from xA to xB. Applying energy conservation KB = KA + W gives us the speed at B.

By performing the calculations, which involve finding the difference in kinetic energy and solving for vB, the student will find the speed of the particle at B.