# Speed, Velocity and Acceleration

The branch of Physics that deals with the motion of a body due to the application of force is called **mechanics**. Mechanics is divided into two branches namely **dynamics** and **statics**.

The branch of mechanics that deals with the state of rest of a body is called statics. The branch of mechanics that deals with the state of motion of a body is called dynamics. Dynamics is classified into kinematics and kinetics.

**Kinematics** is the study of motion which relates to the motion of bodies without reference to either the mass or the force causing it. **Kinetics** is the study of motion which relates to the action of forces causing the motion and the mass that is moved.

A body is said to be at **rest **if its position remains constant with respect to its surroundings or frame of reference. A body is said to be in **motion** if its position is changing with respect to its surroundings or frame of reference.

The line joining the successive positions of a moving body is called its path. The length of the path gives the **distance** travelled by the body. **Displacement** is the directed line segment joining the initial and final positions of a moving body. It is a vector.

If every particle of a moving body traverses the same distance along parallel paths, which may be straight or curved, while the body is moving, then the motion of the body is called **translatory motion**. When the path traversed by each particle of a body is a straight line, then its motion is said to be **rectilinear**. When the path traversed by the particles are parallel paths, then the motion is said to be **curvilinear**.

### Speed

Speed of a body is the rate at which it describes its path. Its SI unit is ms^{-1}. Speed is a scalar quantity.

**Speed = distance travelled / time taken**

A body is said to be moving with **uniform speed** if it has equal distances in equal intervals of time, however small these intervals may be. A body is said to be moving with **non uniform speed** if it has unequal distances in equal intervals of time or equal distances in unequal intervals of time, however small these intervals may be.

**Average speed = total distance travelled / total time taken**

If a particle covers the first half of the total distance with speed a and the second half with speed b,

Average speed = 2ab/(a+b)

If a particle covers 1st 1/3^{rd} of a distance with speed a, 2nd 1/3^{rd} of the distance with speed b and 3rd 1/3^{rd} of the distance with speed c,

Average speed = 3abc/(ab + bc + ca)

1 kmph = 5/18 ms^{-1} ; 1 mph = 22/15 fts^{-1}

For a body with uniform speed, distance travelled = speed x time.

### Velocity

The rate of change of displacement of a body is called velocity. Its SI unit is ms^{-1}. Velocity is a vector quantity.

A body is said to be moving with **uniform velocity**, if it has equal displacements in equal intervals of time, however small these intervals may be. For a body moving with uniform velocity, the displacement is directly proportional to the time interval.

If the direction or magnitude or both of the velocity of a body change, then the body is said to be moving with **non-uniform velocity**.

Average velocity = net displacement / total time taken

For a body moving with uniform acceleration, the average velocity = (u + v)/2

### Acceleration

If the velocity of a body changes **either in magnitude or in direction** or both, then it is said to have acceleration.

For a freely falling body, the velocity changes in magnitude and hence it has acceleration. For a body moving round a circular path with a uniform speed, the velocity changes in direction and hence it has acceleration. For a projectile, whose trajectory is a parabola, the velocity changes in magnitude and in direction, and hence it has acceleration.

The acceleration and velocity of a body need not be in the same direction. For example, a body thrown vertically upwards.

If equal changes of velocity takes place in equal intervals of time, however small these intervals may be, then the body is said to be in **uniform acceleration**. Negative acceleration is called retardation or **deceleration**.

For a body moving with uniform acceleration, the average velocity = (u + v)/2

A body can have zero velocity and non-zero acceleration. For example, for a particle projected vertically up, velocity at the highest point is zero, but acceleration is -g.

If a body has a uniform speed, it may have acceleration. For example, uniform circular motion. If a body has uniform velocity, it has no acceleration.

When a body moves with uniform acceleration along a straight line and has a distance x travelled in the n^{th} second, in the next second it travels a distance x+a, where a is the acceleration.

Acceleration of free fall in vacuum is uniform and is called **acceleration due to gravity** (g) and it is equal to 9.8 ms^{-2}.

**Displacement-time Graph**

- Straight lines represents uniform velocity.
- Slope of straight line gives velocity.
- Smooth curves represents uniform acceleration.
- Zig zag curve represents non-uniform acceleration.

**Velocity-time Graph**

- Slope gives the acceleration.
- Area under the graph gives the distance travelled.
- Curve represents non-uniform acceleration.
- Straight line represents uniform acceleration.

The **equations of motion** for uniform acceleration:

- v = u + at
- s = ut + ½at
^{2} - v
^{2}- u^{2}= 2as - s
_{n}= u + a/2(2n - 1) - s = (u + v)/2 x t