A ball is projected from the ground at an angle of $45°$ with the horizontal surface. It reaches a maximum height of $120 \mathrm{m}$ and returns to the ground. Upon hitting the ground for the first time, it loses half of its kinetic energy. Immediately after the bounce, the velocity of the ball makes an angle of $30°$ with the horizontal surface. The maximum height it reaches after the bounce, in metres, is ___.

A ball is thrown from ground at an angle $\theta$ with horizontal and with an initial speed $u_0$. For the resulting projectile motion, the magnitude of average velocity of the ball up to the point when it hits the ground for the first time is $V_1$. After hitting the ground, the ball rebounds at the same angle $\mathrm{\theta }$ but with a reduced speed of $\frac{u_0}{\alpha }$. Its motion continues for a long time as shown in figure. If the magnitude of average velocity of the ball for entire duration of motion is $0.8V_1$, the value of $\mathrm{\alpha }$ is,

A particle has an initial velocity $3\hat{\mathrm{i}}+4\hat{\mathrm{j}}$ and an acceleration of $0.4\hat{\mathrm{i}}+0.3\hat{\mathrm{j}}$. Its speed after $10 \mathrm{s}$ is,

A plane is inclined at an angle $\mathrm{\alpha }=30°$ with respect to the horizontal. A particle is projected with a speed $u=2 \mathrm{m} {\mathrm{s}}^{-1}$ from the base of the plane making an angle $\mathrm{\theta }=15°$ with respect to the plane as shown in the figure. The distance from the base at which the particle hits the plane is close to:
(Take $g=10 \mathrm{m} {\mathrm{s}}^{-2}$ )