Question

The net force acting to a 2.0-kg discuss while it is being thrown is (αt^2)i + (β+γt)j where α =25.0N/s^2, β = 30.0N and γ = 5.0N/s. If the discuss was originally at rest, what is the velocity after the net force has acted for 0.500 s? Express your answers in terms of the unit vectors i and j.
i- points in the positive x-axis direction
j- points in the positive y-axis direction

Answer 1

for 0.500 s? Express your answers in terms of the unit vectors i and j.

i- points in the positive x-axis direction, j- points in the positive y-axis direction.

$$F(t) = (\alpha t^2)\vec{i} + (\beta + \gamma t)\vec{j} = m\vec{a} \rightarrow \vec{a} = \frac{(\alpha t^2)\vec{i} + (\beta + \gamma t)\vec{j}}{m}$$ $$\vec{v} = \int_0^T \vec{a} \, dt = \int_0^5 \frac{(\alpha t^2)\vec{i} + (\beta + \gamma t)\vec{j}}{m} \, dt = \frac{1}{m} \left( \frac{\alpha 0.5^3}{3} \right) \vec{i} + \left( \beta 0.5 + \frac{\gamma 0.5^2}{2} \right) \vec{j}$$

i- points in the positive x-axis direction

$$v_i = \frac{1}{2} \cdot 25 \cdot \frac{0.0625}{3} = 0.26 \frac{m}{s}$$ $$v_j = 30 \cdot 0.5 + 5 \cdot \frac{0.25}{2} = 15.625 \frac{m}{s}$$