An object is projected at an angle of $45^\circ$ with the horizontal. The ratio of the horizontal range to the maximum height reached is:

The initial velocity of a projectile is given by $v = a\hat{i} + b\hat{j}$. If the range $R$ is twice the maximum height $H$,then which of the following is true?

$A$ player kicks a football with an initial speed of $25\, ms^{-1}$ at an angle of $45^{\circ}$ from the ground. What are the maximum height and the time taken by the football to reach the highest point during motion? (Take $g = 10\, ms^{-2}$)

The initial speed of a bullet fired from a rifle is $630 \; m/s$. The rifle is fired at the center of a target $700 \; m$ away at the same level as the target. How far above the center of the target (in $m$) must the rifle be aimed in order to hit the target? (Take $g = 10 \; m/s^2$)

At what point of a projectile motion are acceleration and velocity perpendicular to each other?

$A$ projectile is given an initial velocity of $(\hat i + 2\hat j) \ ms^{-1}$,where $\hat i$ is along the ground and $\hat j$ is along the vertical. If $g = 10 \ m/s^2$,the equation of its trajectory is