Introduction
If a body is in uniform motion (velocity = constant or body has acceleration and velocity acting at 0° or 180°, then path followed is straight line and motion is rectilinear motion.
Uniform motion
Velocity is constant and acceleration is zero.
Body covers equal displacement in equal time interval.
Equation for uniform motion s = ut .
time-displacement graph is straight line .
Time velocity graph is straight line parallel to time axis
Time acceleration graph is straight line coincide it to time axis.
Variable motion at constant acceleration
Velocity changes at constant rate and acceleration is constant.
Body may cover equal distances in equal time intervals but never covers equal displacements in equal time interval (a and v at 180°)
Distance and Displacement
The length of actual path followed by a moving body is called distance. It is a scalar quantity.
The shortest distance between initial and final position with direction for a moving body is called displacement . It is a vector quantity.
Displacement covered by a moving body in certain time interval may be zero, negative or positive but distance never be zero or negative.
Distance ≥ Displacement
For a moving body, the distance travelled always increases with time displacement may increase or decrease with time.
Speed and Velocity
- Speed
The time rate of change in distance for a moving body is called speed. It is a scalar quantity.
Speed =
The speed of a body can be zero or positive but never negative.
A body is said to be moving with uniform speed if it covers equal distances in equal intervals of time.
A body is said to be moving with a variable speed if it covers equal distances in unequal intervals of time or unequal distances in equal intervals of time.
Average speed =
Instantaneous speed is limiting value of average speed as time interval tends to zero.
It is defined for a point of time .
Velocity
The time rate of change in displacement tor a moving body is called velocity . It is a vector quantity .
Velocity=
Average velocity =
It is defined for period of time.
Instantaneous speed is limiting value of average speed as time interval tends to zero.
It is defined for a point of time .
Average velocity and Average speed
Average velocity for a moving body in certain time interval may be zero, +ve or -ve but average speed can never be zero or –ve .
Average speed ≥ Average velocity
Average velocity
When a body moving along straight line covers distances at different velocities then
Average velocity =
Equal Distance
If when body covers equal distances then ,
Average velocity is harmonic mean of individual velocities .
For two equal distance
Equal Time
If when body covers equal distances then ,
For two equal time.
When a moving body covers two equal parts of distances with two different constant velocities and in same or any directions, then
Average speed (v) is given by
But average velocity is given by
When a moving body travels at two different constant velocities and for equal intervals of time, then Average speed =
Acceleration
The rate of change of velocity with respect to time is called acceleration. It is a vector quantity.
Acceleration=
Average acceleration =
It is defined for a period of time.
Instantaneous acceleration is limiting the value of average speed as the time interval tends to zero.
It is defined for a point of time.
Acceleration
If the change in velocity is constant with time. It is either increasing or decreasing by an equal amount in equal interval of time. It is a uniform acceleration.
If the change in velocity is different with time. It is either increasing or decreasing by an unequal amount in equal interval of time. It is a non-uniform acceleration.
If
velocity(v) =
acceleration(a) =
If
displacement(s) =
acceleration(a) =
If
velocity(v) =
displacement(s) =
Motion with Constant Acceleration
When the motion of the body starts from rest and moves with constant acceleration 'a'. The distance traveled and velocity in nth second is given by:
The distance traveled in successive equal interval of time are in the ratio of
The distance after the end of the successive equal interval of time are in the ratio of
The velocity after the end of the successive equal interval of time are in the ratio of
The time taken to reach the end of successive equal intervals of time are in the ratio of
The time taken to cover successive equal intervals of time are in the ratio of
When the motion of the body starts from rest and moves with constant acceleration 'a'. If be the velocities at successive points at equal separation x, then
When the motion of the body starts from rest and moves with constant acceleration 'a'. If be the velocities at equal time interval t, then
When a particle accelerates from rest for time over distance and at the acceleration and then retards to rest at the rate of over distance in time , then
Net acceleration, (a) =
Maximum velocity () =
Average velocity
Motion Under Gravity
A body is released from 'H' under the influence of gravity takes time t to reach the ground. Then,
The time taken to reach the ground is
The velocity with which it hits ground is
The distance covered in last second
The velocity of body 'h' height above ground
The average velocity for entire motion
The distance traveled in successive equal interval of time are in the ratio of
The distance after the end of the successive equal interval of time are in the ratio of
When a body is thrown vertically upward with initial velocity u under gravity and returns to the thrower's hand. then
Maximum height attached:
Time taken to reach the maximum height
Time of flight
Velocity of the body at height 'h' above the ground
Average speed for entire trip in upward motion:
Distance covered in last second of upward motion is:
The body passes a point at height twice after time and from starting then
Time to reach maximum height
Total time of flight
Height of that point from the ground,
Maximum height attained,
Initial velocity,
When a body of mass 'm' is released from rest from height 'h' along a smooth inclined plane having inclination angle and length 'l', then
Velocity of the body at the bottom of inclined plane
Acceleration down the plane
Time taken to fall down the plane
Time taken to slide down along different smooth inclined planes of different inclination from same height are in ratio
Time taken to slide down along different smooth inclined planes of equal length of different inclination are in the ratio
Relative Velocity
- Introduction
Let velocity of body A be and the velocity of body B be .