A train of mass M is in motion. When its velocity is `u_1` and acceleration is `a_1` , resistive force against its speed is `R_1` . Also when velocity is `u_2`and acceleration is `a_2`, resistive forced is `R_2` . If the engine works at a fixed rate P, show that
`P(u_2-u_1)=u_1u_2(R_1-R_2)+Mu_1u_2(a_1-a_2)`.

A particle starts with the velocity u and moves in a straight line, its acceleration being always equal to its displacement. If v be the velocity when its displacement is x, then show that v^2= u^2+x^2 .

A particle starts with the veloctiy u and moves in a straight line, its acceleration being always equal to its displacement. If v be the velocity when its displacement is x, the show that v^(2)=u^(2)+x^(2) .

Two particles of masses m_ 1, m_2 move with initial velocities u_1 and u_2 . On collision , one of the particles get excited to higher level , after , absorbing energy varepsilon . If final veloctites of particles be v_1 and v_2 , then we must have

A body is projected vertically upwards with an initial velocity u_1 . Another is projected with initial velocity u_2 at an angle theta with the horizontal . If both of them reach the same height ,show that sin theta=(u_1)/(u_2) .

If the area of the circle is A_1 and the area of the regular pentagon inscribed in the circle is A_2, then find the ratio (A_1)/(A_2)dot

If u_n=sin^("n")theta+cos^ntheta, then prove that (u_4-u_6)/(u_2-u_4)=1/2 .

During the n th second of its motion a body covers a distance s_(n) with uniform acceleration a and initial velocity u. Show that a = (2s_(n)-2u)/(2n-1) .

Two trains, moving at velocities u_(1) and u_(2) travel along the same line towards each other. Drivers of both the trains when the trains are at a distance x apart simultaneously apply brakes producing retardations a_(1) and a_(2) , respectively. Prove that a collision can be avoided only if u_(1)^(2)a_(2)+u_(2)^(2)a_(1)=2a_(1)a_(2)x.