Derive the equation, s = u t + 1 2 a t 2 .

The velocity-time graph for a body under uniform acceleration is shown in the figure given below: Let, initial velocity of the body = u Final velocity of the body = v Time taken by the body = t Acceleration of the body = a For a body under uniform acceleration, one can write, Distance , s = Area of the trapezium O A D C or , s = 1 2 × base × height + length × breadth s = 1 2 × A D × B D + O C × O A . . . . ( 1 ) From the figure, A D = O C = t B D = B C - D C = v - u = u + a t - u = a t O A = u Substituting for these in equation ( 1 ) , s = 1 2 × t × a t + t + u s = u t + 1 2 a t 2 This is the second equation of motion.

Derive the relationship, F = m v - u t , where each symbol has its usual significance.

Let a force F acting on a body of mass m produces an acceleration a in the body. Then from the second law of motion, F = m × a . . . . . ( 1 ) If a force F , acting on the body changes its velocity from u (initial velocity) to v (final velocity) in time t , then acceleration is given by the following formula: a = v - u t . . . . ( 2 ) Substituting the value of a in equation ( 1 ) , we get: F = m × a = m v - u t