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

Two particles of masses m₁ and m₂ in projectile motion have velocities ${\overrightarrow{v}}_1$ and ${\overrightarrow{v}}_2$ respectively at time t = 0. They collide at time t₀. Their velocities become ${\overrightarrow{v}}_1\text{'}$ and ${\overrightarrow{v}}_2\text{'}$ at time 2t₀ while still moving in air. The value of $\left|(m_1\overrightarrow{v_1}\text{'}+m_2\overrightarrow{v_2}\text{'})-(m_1\overrightarrow{v_1}+m_2\overrightarrow{v_2})\right|$ is 

1. Zero

2. $(m_1+m_2)g{t}_0$

3. $2(m_1+m_2)g{t}_0$

4. $\frac{1}{2}(m_1+m_2)g{t}_0$

Consider elastic collision of a particle of mass m moving with a velocity u with another particle of the same mass at rest. After the collision the projectile and the struck particle move in directions making angles θ₁ and θ₂ respectively with the initial direction of motion. The sum of the angles. θ₁ + θ₂, is 

(1) 45°

(2) 90°

(3) 135°

(4) 180°

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

The pointer reading v/s load graph for a spring balance is as given in the figure. The spring constant is-

(1) 0.1 N/cm

(2) 5 N/cm

(3) 0.3 N/cm

(4) 1 N/cm

Adjacent figure shows the force-displacement graph of a moving body, the work done in displacing body from x = 0 to x = 35 m is equal to-

(1) 50 J

(2) 25 J

(3) 287.5 J

(4) 200 J

A 10kg mass moves along x-axis. Its acceleration as a function of its position is shown in the figure. What is the total work done on the mass by the force as the mass moves from x = 0 to x = 8 cm ?

(1) 8 × 10^–2 joules

(2) 16 × 10^–2 joules

(3) 4 × 10^–4 joules

(4) 1.6 × 10^–3 joules