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A $4 \mathrm{~kg}$ object has a velocity, $3.0 \hat{\mathrm{i}} \mathrm{m} / \mathrm{s}$ at some instant. Eights seconds later, its velocity is $(8.0 \hat{\mathbf{i}}+10.0 \hat{\mathbf{j}}) \mathrm{m} / \mathrm{s}$. Assuming that the object is subjected to a constant net force, the magnitude of the force is
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2887 Upvotes
Verified Answer
The correct answer is:
$\frac{5 \sqrt{5}}{2} \mathrm{~N}$
Given, mass of an object, $m=4 \mathrm{~kg}, \mathrm{u}=3 \mathrm{im} / \mathrm{s}$ $\mathbf{v}=(8 \hat{\mathbf{i}}+10 \hat{\mathrm{j}}) \mathrm{m} / \mathrm{s}$ and time $t=8 \mathrm{~s}$
As we know, equation of the motion
$$
\begin{array}{rlrl}
\mathbf{v} & =\mathbf{u}+\mathbf{a} t \\
\Rightarrow & & 8 \hat{\hat{\mathbf{i}}}+10 \hat{\mathbf{j}} & =3 \hat{\hat{\mathbf{i}}}+\mathbf{a} \times 8 \\
\Rightarrow & & \mathbf{a} & =\frac{1}{8}(5 \hat{\mathbf{i}}+10 \hat{\mathrm{j}}) \mathrm{m} / \mathrm{s}^2
\end{array}
$$
So, force $\mathbf{F}=$ ma
$$
\begin{aligned}
\quad \mathbf{F} & =\frac{4}{8}(5 \hat{\mathrm{i}}+10 \hat{\mathrm{j}})=\frac{1}{2}(5 \hat{\mathrm{i}}+10 \hat{\mathrm{j}}) \mathrm{N} \\
\Rightarrow \quad F & =\frac{1}{2} \sqrt{5^2+10^2}=\frac{5 \sqrt{5}}{2} \mathrm{~N}
\end{aligned}
$$
Hence, the option (1) is correct.
As we know, equation of the motion
$$
\begin{array}{rlrl}
\mathbf{v} & =\mathbf{u}+\mathbf{a} t \\
\Rightarrow & & 8 \hat{\hat{\mathbf{i}}}+10 \hat{\mathbf{j}} & =3 \hat{\hat{\mathbf{i}}}+\mathbf{a} \times 8 \\
\Rightarrow & & \mathbf{a} & =\frac{1}{8}(5 \hat{\mathbf{i}}+10 \hat{\mathrm{j}}) \mathrm{m} / \mathrm{s}^2
\end{array}
$$
So, force $\mathbf{F}=$ ma
$$
\begin{aligned}
\quad \mathbf{F} & =\frac{4}{8}(5 \hat{\mathrm{i}}+10 \hat{\mathrm{j}})=\frac{1}{2}(5 \hat{\mathrm{i}}+10 \hat{\mathrm{j}}) \mathrm{N} \\
\Rightarrow \quad F & =\frac{1}{2} \sqrt{5^2+10^2}=\frac{5 \sqrt{5}}{2} \mathrm{~N}
\end{aligned}
$$
Hence, the option (1) is correct.
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