Answer
$F=8.36N$
Work Step by Step
Remember that according to Newton's second law, $$F(t)=ma(t)$$ Also note that $$a(t)=\frac{d^2}{dt^2}x(t)$$ Therefore, find the acceleration of each component using differentiation and power rule. $$x'(t)=2.00-12.0t^2$$ $$x''(t)=-24.0t$$ $$y'(t)=7.00-18.0t$$ $$y''(t)=18.0$$ Remember that $$\sqrt{A_x^2+A_y^2}=A^2$$ Use $t=0.700s$ for both equations to get $$x''(0.700)=a_x(0.700)=-24.0(.700)=-16.8m/s^2$$ $$y''(0.700)=a_y(0.700)=18.0m/s^2$$ Use the equation to get an acceleration of $$a=\sqrt{(18.0m/s^2)^2+(-16.8m/s^2)^2}$$ $$a=24.6m/s^2$$ Remember that $F=ma$, so using the acceleration and $m=0.340kg$ yields a force of $$F=(0.340kg)(24.6m/s^2)=8.36N$$
