Question

A single, nonconstant force acts in the + x ‑direction on an object of mass M that is constrained to move along the x ‑axis. As a result, the object's position as a function of time is x ( t ) = P + Q t + R t 3 How much work W is done by this force from t = 0 s to final time T ? Express the answer in terms of P , Q , R , M , and T .

Answer 1

Answer:

Velocity  $V = \frac{dx}{dt} = \frac{d}{dt} (P + Qt + Rt^{3})$

               $V = 0 + Q(1) + 3Rt^{2}$

               V  = Q + 3Rt²

at t = 0, $v_{i}$ = Q + 3r(0) ==> Q

at t = T, $v_{f}$ = Q + 3rT²

Work done (W) = ΔKE = $\frac{1}{2} m(v_f^{2} - v_i^{2})$

W = $\frac{1}{2} m[(Q + 3RT^{2})^{2} - Q^{2}]$

W =  $\frac{1}{2} m$ [Q² + 9R²T⁴ + 2Q(3RT²) - Q²]

W =  $\frac{1}{2} m$ (9R²T⁴ + 6QRT²)

Explanation:

Differentiate the position.

Find the equation for speed.

Find the initial and final speed.

Use work energy theorem to find the work done by finding the change in kinetic energy.