Which one of the following statements is not true?
1. | The magnitude of a vector is always a scalar. |
2. | The average speed of a particle is either greater or equal to the magnitude of the average velocity of the particle over the same interval of time. |
3. | Three vectors not lying in a plane can never add up to give a null vector. |
4. | Each component of a vector is always a scalar. |
A cyclist starts from the center \(\mathrm{O}\) of a circular park of radius \(1\) km, reaches the edge \(\mathrm{P}\) of the park, then cycles along the circumference, and returns to the center along \(\mathrm{QO}\) as shown in the figure. If the round trip takes \(10\) min, then the average speed of the cyclist is:
1. \(22.42\) km/h
2. \(23.32\) km/h
3. \(21.42\) km/h
4. \(27.12\) km/h
The position of a particle is given by,
\(\overrightarrow{\text{r}}=(3.0 \text{t} \hat{\text{i}}-2.0 \text{t}^2 \hat{\text{j}}+4.0 \hat{\text{k}} )~\text{m}\)
where \(t\) is in seconds and the coefficients have the proper units for \(\vec{r}\) to be in meters. What is the magnitude and direction of the velocity of the particle at \(t=2.0\) s?
1. \(7.56~ \text{m} \text{s}^{-1},-70^{\circ}\text{ with} ~\text{y} \text{-axis}. \)
2. \(7.56~ \text{m} \text{s}^{-1}, ~70^{\circ}\text{ with} ~\text{x} \text{-axis}. \)
3. \(8.54 ~\text{m} \text{s}^{-1},~70^{\circ}\text{ with} ~\text{y} \text{-axis}. \)
4. \(8.54 ~\text{m} \text{s}^{-1},-70^{\circ}\text{ with} ~\text{x} \text{-axis}. \)
1. | \(\vec{v}_{\text {avg }}=\frac{1}{2}\left[\vec{v}\left(t_1\right)+\vec{v}\left(t_2\right)\right]\) |
2. | \(\vec{v}(t)=\vec{v}(0)+\vec{a} t\) |
3. | \(\vec{r}({t})=\vec{r}(0)+\vec{v}(0){t}+\frac{1}{2} \vec{a}{t}^2\) |
4. | \(\vec{v}_{\text {avg }}=\frac{\left[\vec{r}\left(t_2\right)-\vec{r}\left(t_1\right)\right]}{\left(t_2-t_1\right)}\) |
An aircraft is flying with speed \(v\) at a height of \(3400~\text{m}\) above the ground (As shown in the figure). If the angle subtended at a ground observation point by the aircraft positions \(10.0~\text{s}\) apart is \(30^{\circ}\), what is the speed of the aircraft? ( Take \(\tan15^{\circ} = 0.267\))
1. \(182~\text{m/s}\)
2. \(130~\text{m/s}\)
3. \(192~\text{m/s}\)
4. \(179~\text{m/s}\)