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Step-by-Step Solution
Step 1: Express the Constant Power in Terms of Force and Velocity
We are given that the car delivers a constant power $P_0$. Power in physics is defined as:
$$P = F \cdot v,$$
where $F$ is the net force acting on the car and $v$ is the velocity. For a car of mass $m$ accelerating at $a$, the force is:
$$F = m\,a.$$
Hence,
$$P_0 = m\,a \cdot v.$$
Step 2: Substitute Acceleration in Terms of Velocity
The acceleration $a$ can be written as the time derivative of velocity:
$$a = \frac{dv}{dt}.$$
Thus,
$$P_0 = m \,\frac{dv}{dt} \,v.$$
Rearrange to separate the differentials:
$$P_0 \, dt = m\,v \, dv.$$
Step 3: Integrate Both Sides
We now integrate both sides to find the relationship between velocity and time. Integrate with respect to $t$ on the left and with respect to $v$ on the right:
$$\int P_0 \, dt = \int m\,v \, dv.$$
Carrying out the integrations:
$$P_0 \, t = \frac{m\,v^2}{2} + C.$$
Because the car starts from rest at $t = 0$, its initial velocity $v(0) = 0$. This implies the integration constant $C = 0.$ Thus,
$$P_0 \, t = \frac{m\,v^2}{2}.$$
Step 4: Solve for the Velocity
Rearranging for $v$:
$$v^2 = \frac{2\,P_0\,t}{m},$$
or
$$v = \sqrt{\frac{2\,P_0\,t}{m}}.$$
Step 5: Conclude the Proportionality
From the final expression:
$$v = \sqrt{\frac{2\,P_0\,t}{m}},$$
we see that $v \propto \sqrt{t}.$ Hence, the instantaneous velocity of the car is proportional to $t^{1/2}.$
