It is a uniform circular motion. So the direction of velocity changes but not the magnitude. Therefore speed in considered constant. Again magnitude of acceleration does not change.

Mass, length are scalars. Acceleration is a vector but magnitude of acceleration is a scalar

Unit vector specifies only direction
\(\hat{A}=\frac{\vec{A}}{|\vec{A}|} \)
\(\therefore \hat{i}+\hat{j}=\frac{\hat{l}+\hat{j}}{|\hat{l}+\hat{j}|} \)
\(\hat{i} \text { and } \hat{j} \text { are orthogonal components of vectors. }\)
\(\therefore|\hat{i}+\hat{j}|=\sqrt{1^{2}+1^{2}}=\sqrt{1+1}=\sqrt{2}\)

\(\text { Gravitational constant } G=\frac{F r^{2}}{m_{l} m_{2}}\)
\(\text { Dimensional formula of } \mathrm{G}=\frac{\left[\mathrm{MLT}^{-2}\right]\left[\mathrm{L}^{2}\right]}{[\mathrm{M}][\mathrm{M}]}\)
\(=\frac{\mathrm{ML}^{3} \mathrm{~T}^{-2}}{\mathrm{M}^{2}}=\mathrm{M}^{-1} \mathrm{~L}^{3} \mathrm{~T}^{-2}\)

\(\text { Torque }=\text { Force } \times \mathrm{r}\)
\(\text { Dimension of torque }\)
\(=\left[\mathrm{MLT}^{-2}\right][\mathrm{L}]=\left[\mathrm{ML}^{2} \mathrm{~T}^{-2}\right]\)
\(\text { Energy }=\text { Force } \times \text { displacement }\)
\(\text { Dimension of Energy }\)
\(=\left[\mathrm{MLT}^{-2}\right][\mathrm{L}]=\left[\mathrm{ML}^{2} \mathrm{~T}^{-2}\right]\)

(i) Displacement is the difference between the final and initial positions of the object in a given interval of time. It can also be defined as the shortest distance between these two positions of the object and its direction is from the initial to final position of the object, during the given interval of time. It is a vector quantity.
(ii) Distance is the actual path length travelled by an object in the given interval of time during the motion. It is a positive scalar quantity.

It is defined as another vector having a magnitude equal to the product of the magnitudes of two vectors and the sine of the angle between them.
If \(\overrightarrow { A } \) and \(\overrightarrow { B } \) are two vectors, then their vector product \(\vec{A} \times \vec{B}=\vec{C}=(A B \sin \theta) \hat{n}\).
The direction of the product vector \((\hat{n})\) is perpendicular to the plane containing the two vectors, in accordance with the right hand screw rule or right hand thumb rule.
1. It is not commutative, i.e., \(\vec{A} \times \vec{B} \neq \vec{B} \times \vec{A}. \ But \ \vec{A} \times \vec{B}=-[\vec{B} \times \vec{A}].\)
2. \((\vec{A} \times \vec{B})_{\max }=A B \hat{n}\), when \(\theta=90^{\circ}\) i.e., when \(\overrightarrow { A } \) and \(\overrightarrow { B } \) are orthogonal to each other.
3. \((\vec{A} \times \vec{B})_{\text {min }}=0\), when \(\theta=0^{\circ} \ or \ 180^{\circ}\) i.e., when the vectors are either parallel or antiparallel provided \(\overrightarrow { A } \) and \(\overrightarrow { B } \) are non-zero vectors.
4. The self - vector products of unit vectors are \(\hat{i} \times \hat{i}=\hat{j} \times \hat{j}=\hat{k} \times \hat{k}=0\)
5. In the case of orthogonal unit vectors, \(\hat{i} \times \hat{j}=\hat{k}, \hat{j} \times \hat{k}=i, \hat{k} \times \hat{i}=j\)
6. Torque \(\tau=\vec{r} \times \overrightarrow{\mathrm{F}}\), angular momentum \(\vec{L}=\vec{r} \times \vec{p}\ an \ \vec{V}=\vec{w} \times \vec{r}\) are examples of vector product.

a) 17.2
b) 4.00\(\times\)10⁵
c) 3.7\(\times\)10^-3
d) 124780

When an object is thrown in the air with some initial velocity (NOT just upwards), and then allowed to move under the action of gravity alone, the object is known as a projectile.
Example:
1. An object dropped from window of a moving train.
2. A bullet fired from a rifle.
3. A ball thrown in any direction.

(i) The Distance travelled by an object in motion in a given time is never negative or zero, it is always positive.
(ii) The displacement of an object, in a given time can be positive, zero or negative.
(iii) The displacement of an object can be equal or less than the distance travelled but never greater than distance travelled.
(iv) The distance covered by an object between two positions can have many values, but the displacement between them has only one value (in magnitude).

The rate of change of linear momentum of a body is directly proportional to the external force applied on the body and this change takes place always in the direction of the applied force.
\(F =\frac{\vec{d} p}{d t}=\frac{d(m \vec{v})}{d t}=m \frac{d \vec{v}}{d t}=m \vec{a}\left(\text { as } a=\frac{d \vec{v}}{d t}\right) \)
\(\vec{F} =m \vec{a} \Rightarrow \text { Force }=\text { mass } \times \text { acceleration }\)