# 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 1/3rd of a distance with speed a, 2nd 1/3rd of the distance with speed b and 3rd 1/3rd 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 nth 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:

1. v = u + at
2. s = ut + ½at2
3. v2 - u2 = 2as
4. sn = u + a/2(2n - 1)
5. s = (u + v)/2 x t