A particle moves a distance ‘x’ in time t according to equation, \(x=\left(t+5\right)^{-1}\). Find: the relation between instantaneous acceleration and velocity of the particle.
Ans: \(a\propto v^{\frac{3}{2}}\).
A particle is moving along x-axis. The position of the particle at any instant is given by; \(x=20+0.1t^{2}\). If, x is measured in meters and t is measured is seconds. Find: (i) Average acceleration of the particle between \(t=2Sec\) to \(t=3Sec\). and (ii) Show that; acceleration of the particle is constant.
Ans:(i)\(a_{avg}=0.2m.s^{-2}\). (ii) \(a=\frac{dv}{dt}=0.2m.s^{-2}\).
A body starts from rest and moves with uniform acceleration. Find: the ratio of the distance covered in the \(n^{th}\) second to the distance covered in \(n\)-seconds.
During the \(n^{th}\) seconds of the motion a body travels a distance \(S_{n}\) with uniform acceleration a and initial velocity \(u\). Then Show That: \(a=\frac{\left(2S_{n}-2u\right)}{\left(2n-1\right)}\).
If, a body loses half of its velocity on penetrating \(3cm\) in wooden block, then how much it will penetrate more before coming to a stop?
Ans: \(S^{‘}=1cm\).
A particle starts from rest and moves a distance \(S_{1}\), with constant acceleration in time \(t\). If, the particle travels distance \(S_{2}\) in next \(t\) Sec, then show that \(S_{2}=3S_{1}\).
A particle located at \(x=0\) at \(t=0\), starts moving along the positive x-direction witha velocity of \(v\), that varies as \(v=a\sqrt{x}\). How, the displacement of the particle varies with time.
Ans: \(x\propto t^{2}\).
The acceleration experienced by a moving car after its engine is switched off is given by; \(a=(-)Kv^{3}\), where K is a positive constant. If, \(v_{0}\) is the magnitude of the velocity when the engine is switched off, then find the velocity of the car at time \(t\) after the engine is switched off?
Ans: \(v=\frac{v_{o}}{\sqrt{2v_{o}Kt+1}}\).
A ball is dropped from a bridge \(122.5m\) above a river. After, the ball has been falling for \(2s\), a second ball is thrown straight downward after it. What must be the initial velocity if the ball so that both balls hit the water of river at the same time?
Ans: \(26.1m.s^{-1}\).
A ball is dropped from the top of a tower of height \(h\). If, covers a distance \(\frac{h}{2}\) in last second of its motion. How long does the ball remains in air?
Ans: \(t=\left(2\pm \sqrt{2}\right)sec\).
The relation between the distance \(x\) and time \(t\) is \(t=\alpha x^{2}+\beta x\). Where; \(\alpha\) and \(\beta\) are constants. Show that retardation is \(2\alpha v^{3}\). Where; \(v\) is instantaneous velocity.
