Final answer:
The problem requires integrating the force to get acceleration and then velocity, followed by integrating velocity to get position. The answer given in the question options appears incorrect because it doesn't account for the mass of the car, which affects the equations of motion.
Step-by-step explanation:
To find the position \(\vec{r}(t)\) and velocity \(\vec{v}(t)\) of the electric car as functions of time given the time-dependent force applied to it, we must integrate the force over time to get the velocity and then integrate the velocity over time to get the position.
Since the electric car starts from rest, its initial velocity is zero, and we are given the force components \(F_x(t) = p + nt\) and \(F_y(t) = qt\). The mass cancels out when we divide force by mass (from Newton's second law \(\vec{F} = m\vec{a}\)), giving us the acceleration components as \(a_x(t) = (p/m) + (n/m)t\) and \(a_y(t) = (q/m)t\).
Integrating acceleration over time gives us the velocity components:
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- \(v_x(t) = \int a_x(t) dt = \int ((p/m) + (n/m)t) dt = (p/m)t + (n/2m)t^2 + C_x\)
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- \(v_y(t) = \int a_y(t) dt = \int (q/m)t dt = (q/2m)t^2 + C_y\)
With the initial condition that the velocity at \(t = 0\) is zero, we find that \(C_x = 0\) and \(C_y = 0\). Hence, the equations for velocity become:
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- \(v_x(t) = (p/m)t + (n/2m)t^2\)
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- \(v_y(t) = (q/2m)t^2\)
Finally, we integrate the velocity to find the position components:
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- \(r_x(t) = \int v_x(t) dt = \int ((p/m)t + (n/2m)t^2) dt = (p/2m)t^2 + (n/6m)t^3 + D_x\)
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- \(r_y(t) = \int v_y(t) dt = \int (q/2m)t^2 dt = (q/6m)t^3 + D_y\)
Knowing that the car starts from the origin, we have \(D_x = 0\) and \(D_y = 0\). Therefore, the position vector is:
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- \(r_x(t) = (p/2m)t^2 + (n/6m)t^3\)
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- \(r_y(t) = (q/6m)t^3\)
However, the options given in the problem statement don't match these derivations because the correct answer would actually require knowledge of the mass of the car, which isn't provided.