The mass of a hydrogen molecule is \(3.32 \times 10^{-27}~\text{kg}.\) If \(10^{23}\) hydrogen molecules strike, per second, a fixed wall of area \(2~\text{cm}^2\) at an angle of \(45^\circ\) to the normal, and rebound elastically with a speed of \(10^3~\text{m/s},\) then the pressure on the wall is nearly:
1. \( 2.35 \times 10^3 ~\text{N/m}^2 \)
2. \(4.70 \times 10^3 ~\text{N/m}^2 \)
3. \(2.35 \times 10^2 ~\text{N/m}^2 \)
4. \(4.70 \times 10^2 ~\text{N/m}^2 \)

A small ball of mass \(m\) is thrown upward with velocity \(u\) from the ground. The ball experiences a resistive force \(mkv^2\) where \(v\) is its speed. The maximum height attained by the ball is:
1. \( \frac{1}{2 k} \ln \left(1+\frac{k u^2}{g}\right) \)
2. \( \frac{1}{2 k} \tan ^{-1} \frac{k u^2}{g} \)
3. \( \frac{1}{k} \ln \left(1+\frac{k u^2}{2 g}\right) \)
4. \( \frac{1}{k} \tan ^{-1} \frac{k u^2}{2 g}\)

A spaceship in space sweeps stationary interplanetary dust. As a result, its mass increases at a rate \(\frac{d M(t)}{d t}=b v^2(t)\), where \(v(t)\) is its instantaneous velocity. The instantaneous acceleration of the satellite is:
1. \( -\frac{2 b v^3}{M(t)} \)
2. \( -\frac{b v^3}{2 M(t)} \)
3. \( -b v^3(\mathrm{t}) \)
4. \( -\frac{b v^3}{M(t)}\)

A particle is projected with velocity \(v_0\) along \(x\)-axis. A damping force is acting on the particle which is proportional to the square of the distance from the origin i.e, \(ma=-\alpha x^2\). The distance at which the particle stops:
1. \( \left(\frac{3m v_0^2}{2 \alpha}\right)^{\frac{1}{2}} \)
2. \( \left(\frac{2mv_0}{3 \alpha}\right)^{\frac{1}{3}} \)
3. \( \left(\frac{2m v_0^2}{3 \alpha}\right)^{\frac{1}{2}}\)
4. \( \left(\frac{3 mv_0^2}{2 \alpha}\right)^{\frac{1}{3}} \)