If W₁, W₂ and W₃ represent the work done in moving a particle from A to B along three different paths 1, 2 and 3 respectively (as shown) in the gravitational field of a point mass m, find the correct relation between W₁, W₂ and W₃ 

(1) W₁ > W₂ > W₃

(2) W₁ = W₂ = W₃

(3) W₁ < W₂ < W₃

(4) W₂ > W₁ > W₃

The displacement x of a particle moving in one dimension under the action of a constant force is related to the time t by the equation $t=\sqrt{x}+3$, where x is in meters and t is in seconds. The work done by the force in the first 6 seconds is 

(1) 9 J

(2) 6 J

(3) 0 J

(4) 3 J

A force $\mathit{F}=-k(y\mathit{i}+x\mathit{j})$ (where k is a positive constant) acts on a particle moving in the xy-plane. Starting from the origin, the particle is taken along the positive x-axis to the point (a, 0) and then parallel to the y-axis to the point (a, a). The total work done by the force on the particle is:

1. $-\mathrm{2k}{a}^2$

2. $\mathrm{2k}{a}^2$

3. $-k{a}^2$

4. \(ka^2\)

A lorry and a car moving with the same K.E. are brought to rest by applying the same retarding force, then:

1. Lorry will come to rest in a shorter distance

2. Car will come to rest in a shorter distance

3. Both will come to rest in a same distance

4. None of the above

A particle free to move along the x-axis has potential energy given by $U\left(x\right)=k[1-e^{-x^2}]$ for $-\infty \le x\le +\infty$, where k is a positive constant of appropriate dimensions. Then 

(1) At point away from the origin, the particle is in unstable equilibrium

(2) For any finite non-zero value of x, there is a force directed away from the origin

(3) If its total mechanical energy is k/2, it has its minimum kinetic energy at the origin

(4) For small displacements from x = 0, the motion is simple harmonic

The kinetic energy acquired by a mass m in travelling a certain distance d starting from rest under the action of a constant force is directly proportional to 

(1) $\sqrt{m}$

(2) Independent of m

(3) $1/\sqrt{m}$

(4) m

An open knife edge of mass 'm' is dropped from a height 'h' on a wooden floor. If the blade penetrates upto the depth 'd' into the wood, the average resistance offered by the wood to the knife edge is 

(1) mg

(2) $mg\left(1-\frac{{\displaystyle h}}{{\displaystyle d}}\right)$

(3) $mg\left(1+\frac{{\displaystyle h}}{{\displaystyle d}}\right)$

(4) $mg{\left(1+\frac{{\displaystyle h}}{{\displaystyle d}}\right)}^2$

A body is moving along a straight line by a machine delivering constant power. The distance moved by the body in time t is proportional to 

(1) t^1/2

(2) t^3/4

(3) t^3/2

(4) t²

A batsman hits a sixer and the ball touches the ground outside the cricket field. Which of the following graph describes the variation of the cricket ball's vertical velocity v with time between the time t₁ as it hits the bat and time t₂ when it touches the ground 

(1)

(2)

(3)

(4)

The relationship between force and position is shown in the given figure (in a one-dimensional case). The work done by the force in displacing a body from \(x = 1\) cm to \(x = 5\) cm is:

1.   \(20\) ergs
2.   \(60\) ergs
3.   \(70\) ergs
4.   \(700\) ergs