Question

At t = 0, an object of mass m is at rest at x = 0 on a horizontal, frictionless surface. A horizontal force Fₓ = F₀(1 − t/T), which decreases from F₀ at t = 0 to zero at t = T, is exerted on the object. Find an expression for the object’s velocity.

Answer 1

The expression for velocity is V = mF₀(T−t)/(mT) - F₀/T, where V is the velocity of the object, m is the mass of the object, F₀ is the initial force applied, and T is the time interval over which the force decreases from F₀ to zero.

Step-by-step explanation:

The equation of motion for the object can be described using Newton's second law, which states that the net force acting on an object is equal to the mass of the object multiplied by its acceleration. The net force on the object is given by Fₓ = F₀(1 − t/T), and since the surface is frictionless, the only force acting on the object is the net force. Therefore, we can equate the net force to the mass of the object multiplied by its acceleration:

Fₓ = m * a

Substituting the expression for Fₓ and differentiating with respect to time, we can find the expression for acceleration:

a = dV/dt = dV/dt * dT/dT = F₀(1−t/T)*m/dT/dt

Integrating the expression for acceleration with respect to time, we can find the expression for velocity:

V = ∫ a dt = ∫ F₀(1−t/T)*m/dT/dt dt = mF₀(T−t)/(mT) + C

where C is the constant of integration. At t = 0, the object is at rest, so the initial velocity (V₀) is zero. Substituting this into the expression for velocity, we can solve for C:

V₀ = mF₀(T−0)/(mT) + C = 0

Simplifying, we find that C = -F₀/T. Substituting this into the expression for velocity, we obtain the final expression:

V = mF₀(T−t)/(mT) - F₀/T