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NEET 2023
Exam
NEET
Year
2023
Subject
Physics
Chapter
Motion in One Dimension

NEET 2023 · Physics · Motion in One Dimension

The position of a particle is given by r (t)=4 t i +2 t^2 j +5 k...

The position of a particle is given by r⃗(t)=4ti^+2t2j^+5k^\vec{r}(t)=4 t \hat{i}+2 t^2 \hat{j}+5 \hat{k} where tt is in seconds and rr in metre. Find the magnitude and direction of velocity v(t)v(t), at t=1 st=1 \mathrm{~s}, with respect to xx-axis.

Options

  1. A32 ms−1,30∘3 \sqrt{2} \mathrm{~ms}^{-1}, 30^{\circ}
  2. B32 ms−1,45∘3 \sqrt{2} \mathrm{~ms}^{-1}, 45^{\circ}
  3. C42 ms−1,45∘4 \sqrt{2} \mathrm{~ms}^{-1}, 45^{\circ}
  4. D42 ms−1,60∘4 \sqrt{2} \mathrm{~ms}^{-1}, 60^{\circ}

Correct answer

C. 42 ms−1,45∘4 \sqrt{2} \mathrm{~ms}^{-1}, 45^{\circ}

Step-by-step solution

r⃗(t)=(4ti^+2t2j^+5k^)mdr⃗(t)dt=4i^+4tj^∣t=1 ms−1v⃗=4i^+4j^ ms−1∣v⃗∣=42+42=42 ms−1tan⁡ϕ=vyvx=44=1ϕ=45∘\begin{aligned} & \vec{r}(t)=\left(4 t \hat{i}+2 t^2 \hat{j}+5 \hat{k}\right) \mathrm{m} \\ & \frac{d \vec{r}(t)}{d t}=4 \hat{i}+\left.4 t \hat{j}\right|_{t=1} \mathrm{~ms}^{-1} \\ & \vec{v}=4 \hat{i}+4 \hat{j} \mathrm{~ms}^{-1} \\ & |\vec{v}|=\sqrt{4^2+4^2}=4 \sqrt{2} \mathrm{~ms}^{-1} \\ & \tan \phi=\frac{v_y}{v_x}=\frac{4}{4}=1 \\ & \phi=45^{\circ}\end{aligned}
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