The position coordinates of a particle in a plane are \(\left(1, 1\right)\) and \(\left(2, 3\right)\) at time \(t_{1}\) and \(t_{2}\). What are the position vectors of the particle at time \(t_{1}\) and \(t_{2}\)? What is the displacement of the particle during this time interval?
Ans: \(|\vec{S}|=\sqrt{5}Units\).
The instantaneous coordinate of a particle are \(x=\left(5t-3\right)m\) and \(y=7t^{2}m\). Calculate: (i) The average velocity of the particle during time interval from \(t=1s\) to \(t=2s\)? (ii) Velocity of the particle at time \(t=2s\).
If, the position vectors of a particle is given by \(\vec{r}=\left[\left(4Cos2t\right)\hat{i}+\left(4Sin2t\right)\hat{j}+\left(6t\right)\hat{k}\right]m\), then what should be the acceleration of the particle at time \(t=\frac{\pi}{4}sec\)?
A particle starts from origin at time \(t=0sec\), with a velocity \(5.0\hat{i} ms^{-1}\) and moves in X-Y plane with a constant acceleration \(\left(2\hat{i}+3\hat{j}\right) ms^{-2}\). Calculate: (i) The time at which X-coordinate of the particle is \(6m\). (ii) Y-coordinate of the particle when X-coordinate of the particle is \(6m\). (iii) Speed of the particle at that time.
Ans:(i) \(t=1s\), (ii) \(y=1.5m\) and (iii) \(|\vec{V}|=7.6\frac{m}{s}\).
The position of a particle is given by: \(\vec{r}=\left(2t\hat{i}+3t^{2}\hat{j}+5\hat{k}\right)m\), where \(t\) is measured in seconds, \(\vec{r}\) in meter (m). Calculate: (i) Velocity and acceleration of the particle. (ii) Magnitude of velocity at time \(t=5s\). and (iii) Direction of velocity of the particle at \(t=5s\).
Ans:(i) \(\vec{V}=\left(2\hat{i}+6t\hat{j}\right)\frac{m}{s}\), & \(\vec{a}=6\hat{j}frac{m}{s^{2}}\) (ii) \(30.07\frac{m}{s}\) and (iii) \(\theta=86^{o}\).
A particle starts from origin at \(t=0\) with a velocity \(\left(5.0\hat{i}\right) ms^{-1}\), and moves in X-Y plane under the action of force which produces a constant acceleration of \(\left(3.0\hat{i}+2.0\hat{j}\right) ms^{-2}\). Calculate: (i) Y-coordinate of the particle at the time instant its X-coordinate is \(84m\). (ii) What is the speed of the particle at this time?
Ans:(i) \(Y=36m\) and (ii) \(25.9\frac{m}{s}\).
The position vector of a particle in X-Y plane at \(t= 0\) is given by \(\left(2\hat{i}+4\hat{j}\right)m\). The velocity acquired by the particle in \(2s\) is \(\left(4\hat{i}+3\hat{j}\right) ms^{-1}\). Find: the X and Y coordinate other particle at \(t=2s\).
Ans: \(X=10m\) and \(Y=10m\).
The position of a particle is given by \(\vec{r}=\left(3t\hat{i}+4t^{3}\hat{j}+5\hat{k}\right)m\). Find: (i) The velocity and acceleration of the particle. (ii) The magnitude of the velocity and acceleration of the particle at \(t=5Sec\).
