The position of a particle is given by

\(r = 3.0t \;\hat{i} – 2.0t^{2} \;\hat{j} + 4.0 \hat{k} \; \;m\)

Where t is in seconds and the coefficients have the proper units for r to be in meters.

**(a)** Find the ‘v’ and ‘a’ of the particle?

**(b)** What is the magnitude and direction of the velocity of the particle at t = 2.0 s?

Asked by Abhisek | 1 year ago | 76

**(a) ****The position of the particle is given by:**

\(\vec{r} = 3.0t \;\hat{i} – 2.0t^{2} \;\hat{j} + 4.0 \hat{k}\)

Velocity \(\vec{v}\), of the particle is given as:

\(\\\vec{v} = \dfrac{\vec{dr}}{dt} = \dfrac{d}{dt}\left ( 3.0t \;\hat{i} – 2.0t^{2} \;\hat{j} + 4.0 \hat{k} \right )\\ \\ \vec{v} = 3.0\; \hat{i} – 4.0t \; \hat{j}\)

Acceleration \(\vec{a}\), of the particle is given as:

\(\\\vec{a} = \dfrac{d\vec{v}}{dt} = \dfrac{d}{dt}\left ( 3.0\; \hat{i} – 4.0t \; \hat{j} \right ) \\ \\ \vec{a} = -4.0 \hat{j}\)

8.54 m/s, \(69.45^{\circ}\)below the x – axis

**(b)** we have velocity vector,\(\vec{v} = 3.0\; \hat{i} – 4.0t \; \hat{j}\)

**At t = 2.0 s:**

\(\vec{v} = 3.0\; \hat{i} – 8.0 \; \hat{j}\)

The magnitude of velocity is given by:

\(\left |\vec{v} \right | = \sqrt{3.0^{2} + – 8.0^{2}} = \sqrt{73} = 8.54 \;m/s\)

Direction, \(\theta\) = \( \tan ^{-1} \left ( \dfrac{v_{y}}{v_{x}} \right )\\ \\ = \tan ^{-1} \left ( \dfrac{-8}{3} \right ) = – \tan ^{-1} \left ( 2.667 \right )\\ \\ = 69.45^{\circ}\)

The negative sign indicates that the direction of velocity is below the x – axis.

Answered by Pragya Singh | 1 year ago**(a) **Show that for a projectile the angle between the velocity and the x-axis as a function of time is given by

\(\theta (t)=tan^{-1}(\dfrac{v_{0y-gt}}{v_{ox}})\)

**(b)** Shows that the projection angle θ_{0} for a projectile launched from the origin is given by

\(\theta_{0}=tan^{-1}(\dfrac{4h_{m}}{R})\)

where the symbols have their usual meaning

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