Prove that `x = x_0 + v_0 t + 1/2 at^(2)`.

The velocity of a particle is given by v=u_(0) + g t+ 1/2 at^(2) . If its position is x =0 at t=0 , then what is its displacement after t=1 s ?

Prove that tan^(-1) x + tan ^(-1). 1/x = {{:(pi//2", if " x gt 0),(-pi//2", if "x lt 0 ):} .

Prove that tan^(-1) x + tan^(-1).(1)/(x) = {(pi//2,"if" x gt 0),(-pi//2," if " x lt 0):}

The velocity of a particle is v = v_0 + g t + ft^2 . If its position is x = 0 at t = 0 , then its displacement after unit time (t = 1) is.

Prove that int_0^1 tan^-1((2x-1)/(1+x-x^2))dx=0

Using L M V T prove that tanx > x in (0,pi/2),

A transverse wave y=A sin omega (x/V_(1) -t) is travelling in a medium with speed V_(1) . Plane x = 0 is the boundary of the medium. For x gt 0 there is a different medium in which the wave travels at a different speed V_(2) . Part of the wave is reflected and part is transmitted. For x lt 0 the wave function is described as y_(-)=A_(1) sin omega (x/V_(1)-t)+A_(2) sin omega (x/V_(2)+t) while for x gt 0, y_(+)=A_(3) sin omega(x/V_(2)-t) (a) Using the fact that the wave function must be continuous at x = 0 , show that A_(1) – A_(2) = A_(3) (b) Using the fact that (del y)/(del x) must be continuous at x = 0 , prove that V_(2)/V_(1) A_(1)=A_(3)-A_(2) (c) Show that A_(3)=(2V_(2)A_(1))/(V_(1)+V_(2)) and A_(2)=(V_(1)-V_(2))/(V_(1)+V_(2))A_(1)

The distance x covered in time t by a body having initial velocity v_(0) and havig a constant acceleration a is given by x=v_(0)t+((1)/(2))at^(2) . This result follows from:-

Prove that e^x-e^(-x)-2x >0\ AA\ x > 0. Hence, prove that e^x+e^(-x)\ geqx^2+2\ AA\ x\ geq0.