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Chapter 2: Problem 304

A particle has initial velocity \((2 \hat{1}+3 \hat{j}) \mathrm{ms}^{-1}\) and has acceleration \((\hat{1}+\hat{j}) \mathrm{ms}^{-2}\). Find the velocity of the particle after 2 second. (A) \((3 \hat{1}+5 \hat{j}) \mathrm{ms}^{-1}\) (B) \((4 \hat{i}+5 \hat{\jmath}) \mathrm{ms}^{-1}\) (C) \((3 \hat{1}+2 \hat{j}) \mathrm{ms}^{-1}\) (D) \((5 \hat{1}+4 \hat{j}) \mathrm{ms}^{-1}\)

Short Answer

Expert verified
The velocity of the particle after 2 seconds is \(4\hat{i} + 5\hat{j} \,\text{ms}^{-1}\).

Step by step solution

01

Identify given information

We are given: - Initial velocity: \(\vec{v_0} = 2\hat{i} + 3\hat{j} \,\text{ms}^{-1}\) - Acceleration: \(\vec{a} = \hat{i} + \hat{j} \,\text{ms}^{-2}\) - Time: \(t = 2\,\text{s}\) We need to find the velocity \(\vec{v}\) after 2 seconds.
02

Write the velocity equation for each component

The equation for the velocity of the particle as a function of time is given by: \[ \vec{v}(t) = \vec{v_0} + \vec{a}t \] To find the x and y components of the velocity, we will write the equation separately for each component: - For x-component: \(v_x(t) = v_{0x} + a_x t\) - For y-component: \(v_y(t) = v_{0y} + a_y t\)
03

Substitute given values into the equations

We will now substitute the given values into the equations: - \(v_x(t=2) = 2\hat{i} + (\hat{i})(2)\) - \(v_y(t=2) = 3\hat{j} + (\hat{j})(2)\)
04

Calculate the components of the velocity

Compute the components: - \(v_x(t=2) = 2\hat{i} + 2\hat{i} = 4\hat{i}\) - \(v_y(t=2) = 3\hat{j} + 2\hat{j} = 5\hat{j}\)
05

Combine the components to find the final velocity vector

Combine the components: \[ \vec{v}(t=2) = 4\hat{i} + 5\hat{j} \] So, the velocity of the particle after 2 seconds is \(4\hat{i} + 5\hat{j} \,\text{ms}^{-1}\), which corresponds to the answer (B).

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Most popular questions from this chapter

Equation of a projectile is given by \(\mathrm{y}=\mathrm{Ax}-\mathrm{Bx}^{2}\). Find the range for the particle. (A) \((\mathrm{A} / \mathrm{B})\) (B) \((\mathrm{A} / 4 \mathrm{~B})\) (C) \((\mathrm{A} / 2 \mathrm{~B})\) (D) \((2 \mathrm{~A} / \mathrm{B})\)

The distance travelled by a particle is given by $\mathrm{s}=3+2 \mathrm{t}+5 \mathrm{t}^{2}\( The initial velocity of the particle is \)\ldots$ (A) 2 unit (B) 3 unit (C) 10 unit (D) 5 unit

A car covers one third part of its straight path with speed \(\mathrm{V}_{1}\) and the rest with speed \(\mathrm{V}_{2}\). What is its average speed? (A) $\left[\left(3 \mathrm{v}_{1} \mathrm{v}_{2}\right) /\left(2 \mathrm{v}_{1}+\mathrm{v}_{2}\right)\right]$ (B) $\left[\left(2 \mathrm{v}_{1} \mathrm{v}_{2}\right) /\left(3 \mathrm{v}_{1}+\mathrm{v}_{2}\right)\right]$ (C) $\left[\left(3 \mathrm{v}_{1} \mathrm{v}_{2}\right) /\left(\mathrm{v}_{1}+2 \mathrm{v}_{2}\right)\right]$ (D) $\left[\left(3 \mathrm{v}_{1} \mathrm{v}_{2}\right) /\left(2 \mathrm{v}_{1}+2 \mathrm{v}_{2}\right)\right]$

Find a unit vector perpendicular to both \(\mathrm{A}^{\rightarrow}\) and \(\mathrm{B}^{\rightarrow}\) (A) $\left[\left(\mathrm{A}^{\rightarrow} \cdot \mathrm{B}^{\rightarrow}\right) / \mathrm{AB}\right]$ (B) $\left[\left(\mathrm{A}^{\rightarrow} \times \mathrm{B}^{-}\right) /(\mathrm{AB} \sin \theta)\right]$ (C) $\left[\left(\mathrm{A}^{\rightarrow} \times \mathrm{B}^{\rightarrow}\right) /(\mathrm{AB} \cos \theta)\right]$ (D) $\left[\left(\mathrm{A}^{\rightarrow} \cdot \mathrm{B}^{\rightarrow}\right) /(\mathrm{AB} \sin \theta)\right]$

A balloon is going vertically up with velocity \(12 \mathrm{~m} / \mathrm{s}\). When it is at height of \(65 \mathrm{~m}\) above the ground, it releases a stone. In how much time the stone will reach the ground. (A) \(\sqrt{13} \mathrm{~s}\) (B) \(10 \mathrm{~s}\) (C) \(5 \mathrm{~s}\) (D) \(6 \mathrm{~s}\)
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