Serway & Jewett — Linear Momentum and Collisions: Problem 51

There are (one can say) three coequal theories of motion for a single particle: Newton’s second law, stating that the total force on the particle causes its acceleration; the work–kinetic energy theorem, stating that the total work on the particle causes its change in kinetic energy; and the impulse–momentum theorem, stating that the total impulse on the particle causes its change in momentum. In this problem, you compare predictions of the three theories in one particular case. A 3.00-kg object has velocity \(7.00 \hat{j} \text{ m/s}\). Then, a constant net force \(12.0 \hat{i} \text{ N}\) acts on the object for \(5.00 \text{ s}\). (a) Calculate the object’s final velocity, using the impulse–momentum theorem. (b) Calculate its acceleration from \(\vec{a} = (\vec{v}_f - \vec{v}_i) / \Delta t\). (c) Calculate its acceleration from \(\vec{a} = \sum \vec{F} / m\). (d) Find the object’s vector displacement from \(\Delta \vec{r} = \vec{v}_i t + \frac{1}{2} \vec{a} t^2\). (e) Find the work done on the object from \(W = \vec{F} \cdot \Delta \vec{r}\). (f) Find the final kinetic energy from \(\frac{1}{2} m v_f^2 = \frac{1}{2} m \vec{v}_f \cdot \vec{v}_f\). (g) Find the final kinetic energy from \(\frac{1}{2} m v_i^2 + W\). (h) State the result of comparing the answers to parts (b) and (c), and the answers to parts (f) and (g).