An object with mass 500 g moves along x-axis with speed v = 4√x m/s. The force acting on the object is:

❓ Question

An object with mass 500 g moves along the x-axis with speed

$$v=4\sqrt{x}\text{ m/s}\mathrm{.}$$

The force acting on the object is: ?

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🖼️ Question Image

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✍️ Short Solution

We are given velocity as a function of position, $v(x) = 4\sqrt{x}$, and mass $m = 0.5\ \text{kg}$.

To find the force, recall Newton’s Second Law:

$$F=m\frac{dv}{dt}\mathrm{.}$$

Since velocity is a function of position, apply chain rule:

$$\frac{dv}{dt}=\frac{dv}{dx}\cdot \frac{dx}{dt}=v\frac{dv}{dx}\mathrm{.}$$

Hence,

$$F=mv\frac{dv}{dx}\mathrm{.}$$

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🔹 Step 1: Compute dv/dx

$$v=4\sqrt{x}⟹\frac{dv}{dx}=4\cdot \frac{1}{2}{x}^{-1\mathrm{/}2}=\frac{2}{\sqrt{x}}\mathrm{.}$$

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🔹 Step 2: Substitute values

$$F=m\cdot v\cdot \frac{dv}{dx}=0.5\cdot (4\sqrt{x})\cdot \frac{2}{\sqrt{x}}\mathrm{.}$$

Simplify:

$$F=0.5\cdot 4\cdot 2=4\text{ N}\mathrm{.}$$

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🧮 Image Solution

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✅ Conclusion & Video Solution

✅ Answer:

$$\boxed{{\displaystyle F=4\text{ N}}}$$

📘 Concept Recap:

• When velocity is given as a function of position, force can be computed using

  $$F=mv\frac{dv}{dx}\mathrm{.}$$

• Here, even though $v$ increases with $x$, the resulting force is constant.

• This implies uniform acceleration, which can be verified:

$$a=F\mathrm{/}m=4\mathrm{/}0.5=8\text{ m/s²}\mathrm{.}$$

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