### Introduction To Kinematics Basic Mathematical Concepts

**Recollection Guide**

**1. Scalars and vectors**

*Scalars*are quantities that have only magnitude, such as mass and temperature.*Vectors*are quantities that have both magnitude and direction, such as force and velocity.

**2. Vector addition and subtraction**

- Vectors can be added and subtracted by using the parallelogram law.
- To add two vectors, place them head-to-tail and draw a parallelogram between them. The vector that goes from the tail of the first vector to the head of the second vector is the sum of the two vectors.
- To subtract a vector from another vector, place them head-to-tail and draw a parallelogram between them. The vector that goes from the head of the first vector to the tail of the second vector is the difference of the two vectors.

**3. Dot product and cross product of vectors**

- The
*dot product*of two vectors is a scalar quantity that is equal to the product of the magnitudes of the two vectors multiplied by the cosine of the angle between them. - The
*cross product*of two vectors is a vector quantity that is perpendicular to both of the two vectors and has a magnitude equal to the product of the magnitudes of the two vectors multiplied by the sine of the angle between them.

**4. Unit vectors**

*Unit vectors*are vectors that have a magnitude of 1.- Unit vectors are often used to represent the directions of vectors.

**5. Position, displacement, velocity, and acceleration**

*Position*is the location of an object at a given time.*Displacement*is the change in position of an object over a given time interval.*Velocity*is the rate of change of an object’s position with respect to time.*Acceleration*is the rate of change of an object’s velocity with respect to time.

**6. Equations of motion**

- The
*equations of motion*are three mathematical equations that describe the motion of an object in a straight line. - The first equation of motion is:

```
v = u + at
```

- where
*v*is the final velocity of the object,*u*is the initial velocity of the object,*a*is the acceleration of the object, and*t*is the time interval. - The second equation of motion is:

```
s = ut + 1/2at^2
```

- where
*s*is the displacement of the object,*u*is the initial velocity of the object,*a*is the acceleration of the object, and*t*is the time interval. - The third equation of motion is:

```
v^2 = u^2 + 2as
```

- where
*v*is the final velocity of the object,*u*is the initial velocity of the object,*a*is the acceleration of the object, and*s*is the displacement of the object.

**7. Projectile motion**

*Projectile motion*is the motion of an object that is launched into the air, such as a ball or a rocket.- The equations of motion for projectile motion are:

```
x = u_xt
```

```
y = u_yt - 1/2gt^2
```

- where
*x*is the horizontal displacement of the object,*u_x*is the initial horizontal velocity,*t*is the time interval,*y*is the vertical displacement of the object,*u_y*is the initial velocity of the object,*g*is the acceleration due to gravity, and*t*is the time interval.

**8. Circular Motion**

*Circular motion*is the motion of an object that moves in a circular path, such as a planet orbiting the sun or a car going around a curve.- The equations of motion for circular motion are:

```
v = ωr
```

```
a = ω^2r
```

- where
*v*is the tangential velocity of the object,*ω*is the angular velocity,*r*is the radius of the circular path, and*a*is the centripetal acceleration.

**9. Relative velocity and acceleration**

*Relative velocity*is the velocity of an object relative to another object.*Relative acceleration*is the acceleration of an object relative to another object.

**10. Angular motion**

*Angular motion*is the rotation of an object around a fixed point.- The equations of angular motion are:

```
ω = Δθ / Δt
```

```
α = Δω / Δt
```

- where
*ω*is the angular velocity of the object,*θ*is the angle that the object has rotated through,*t*is the time interval, and*α*is the angular acceleration of the object.

**11. Uniform circular motion**

*Uniform circular motion*is circular motion in which the object moves with a constant angular velocity.- The equations of uniform circular motion are:

```
v = ωr
```

```
a = ω^2r
```

- where
*v*is the tangential velocity of the object,*ω*is the angular velocity,*r*is the radius of the circular path, and*a*is the centripetal acceleration.