For objects in uniformly accelerated rectilinear motion, the five quantities, displacement x, the time took t, initial velocity v0, final velocity v and acceleration a are related by a set of simple equations called kinematic equations of motion :
if the position of the object at time t = 0 is 0. If the particle starts at x = x0 , x in above equations is replaced by (x- x0).
The ‘displacement-time’ and the ‘velocity-time’ graphs of a particle are often used to provide us with a visual representation of the motion of a particle. The ‘shape’ of the graphs depends on the initial ‘co-ordinates’ and the ‘nature’ of the acceleration of the particle (Fig.)
Equations of Uniformly Accelerated Motion
If a body starts with velocity (u) and after time t its velocity changes to v, if the uniform acceleration is a and the distance traveled in time t in s, then the following relations are obtained, which are called equations of uniformly accelerated motion.
- v = u + at
- s = ut + at2
- v2 = u2 + 2as
Distance traveled in nth second
If a body moves with uniform acceleration and velocity changes from u to v in a time interval, then the velocity at the midpoint of its path
√u2 + v2/ 2
Motion under Gravity
If an object is falling freely (u = 0) under gravity, then equations of motion
- v = u + gt
- h =ut +gt2
- v2 = u2 + 2gh
Note: If an object is thrown upward then g is replaced by – g in above three equations. It thus follows that
Time is taken to reach maximum height:
Maximum height reached by the body:
A ball is dropped from a building of height h and it reaches after t seconds on earth. From the same building if two balls are thrown (one upwards and other downwards) with the same velocity u and they reach the earth surface after t, and t2 seconds respectively, then
When a body is dropped freely from the top of the tower and another body is projected horizontally from the same point, both will reach the ground at the same time.