A trebuchet was a hurling machine built to attack the walls of a castle under siege. A large stone could be hurled against a wall to break apart the wall. The machine was not placed near the wall because the arrows could reach it from the castle wall. Instead, it was positioned, so that, the stone hit the wall during the second half of its flight. Suppose a stone is launched with a speed of $v_{\mathrm{B}}=30.0 \mathrm{m} {\mathrm{s}}^{-1}$ and at an angle of ${\text{θ}}_0=40.0^{°}$. What is the speed of the stone if it hits the wall

(a) Just as it reaches the top of its parabolic path and

(b) when it has descended to half that height?

(c) As a percentage, how much faster is it moving in part (b) than in part (a)?

A rifle that shoots bullets at $460 \mathrm{m} {\mathrm{s}}^{-1}$ is to be aimed at a target $45.7 \mathrm{m}$ away. If the center of the target is level with the rifle, how high above the target must the rifle barrel be pointed so that the bullet hits dead center?

During a tennis match, a player serves the ball at $23.6 \mathrm{m} {\mathrm{s}}^{-1}$, with the center of the ball leaving the racquet horizontally $2.42 \mathrm{m}$ above the court surface. The net is $12 \mathrm{m}$ away and $0.90 \mathrm{m}$ high. When the ball reaches the net,

(a) does the ball clear it and

(b) what is the distance between the center of the ball and the top of the net?

Suppose that, instead, the ball is served as before but now it leaves the racquet at $5.{00}^{°}$below the horizontal when the ball reaches the net,

(c) does the ball clear it and

(d) what now is the distance between the center of the ball and the top of the net?

From the platform edge located at $12.0 \mathrm{m}$ above the surface of the water, a high champion pushes off horizontally with a speed of $2.50 \mathrm{m} {\mathrm{s}}^{-1}$.

(a) At what horizontal distance from the edge is the diver $0.900 \mathrm{s}$ after pushing?

(b) At what vertical distance above the surface of the water is the diver just then?

(c) At what horizontal distance from the edge does the diver strike the water?

A golf ball is struck at ground level. The speed of the golf ball as a function of the time is shown in the figure, where, $t=0$ at the instant the ball is struck. The scaling on the vertical axis is set by $v_{\mathrm{a}}=19 \mathrm{m} {\mathrm{s}}^{-1}$ and $v_{\mathrm{b}}=31 \mathrm{m} {\mathrm{s}}^{-1}$.

(a) How far does the golf ball travel horizontally before returning to ground level?

(b) What is the maximum height above ground level attained by the ball?

In the figure below, a ball is thrown leftward from the left edge of the roof, at height $h$ above the ground. The ball hits the ground $1.50 \mathrm{s}$ later, at distance $d=25.0 \mathrm{m}$ from the building and at angle $\mathrm{\theta }=60.0^{°}$ with the horizontal.

(a) Find $h$ (Hint: One way is to reverse the motion, as if on video.)

(b) What is the magnitude?

(c) What is the angle relative to the horizontal of the velocity at which the ball is thrown?

(d) Is the angle above or below the horizontal?

Suppose that a shot putter can put a shot at the world-class speed $v_0=15.00 \mathrm{m} {\mathrm{s}}^{-1}$ and at a height of $2.160 \mathrm{m}$. What horizontal distance would the shot travel if the launch angle ${\mathrm{\theta }}_0$ is 

(a) $45.00°$ and 

(b) $42.00°$? 

Upon spotting an insect on a twig overhanging water, an archerfish squirts water drops at the insects to knock it into the water (figure below). Although the insects is located along a straight-line path at angle $\varphi$ and distance $d$, a drop must be launched at a different angle ${\mathrm{\theta }}_0$ if its parabolic path is to intersect the insect. If $\varphi =36.0°$ and $d=0.900 \mathrm{m}$, what launch angle ${\mathrm{\theta }}_0$ is required for the drop to be at the top of the parabolic path when it reaches the insect?

In $1939$ or $1940$, Emanuel Zacchini took his human-cannonball act to an extreme, After being shot from a cannon, he soared over three Ferris wheels and into a net In figure. Assume that he is launched with a speed of $26.5 \mathrm{m} {\mathrm{s}}^{-1}$ and at an angle of $53.0°$.

$\left(a\right)$ Treating him like a particle calculates his clearance over the first wheel.

$\left(b\right)$ If the reached maximum height over the middle wheel, by how much did he clear it?

$\left(c\right)$ How far from the cannon should the net's center have been positioned (neglect air drag)?