Question

. A van traveling 60 mi/hr sees a duck in the distance and slams on its brakes. If the duck is exactly 3 feet away when the van begins braking, how quickly must it accelerate in order to save c. A duck on a motorcycle is speeding down the highway at 65 mi/hr where the speed limit is 00 the duck? only 55 mi/hr. It passes a police car on the side of the road but does not slow down. The instant the the police, the police car begins to accelerate forward (from rest) to arrest the duck accelerate at a rate of 0.25 g's, how long will it take until the police car catches the duck? duck passes 2. A cannon is set at an angle of 30°, and its muzzle velocity is 125 mi/hr. You are trying to launch a annonball so that it perfectly hits a princess standing on the top of her castle, at a height of 75 feet above the ground. How far back do you have to place the cannon from the castle? A ssume the princess is at the edge of the castle's wall taunting you, and there is no atmosphere in the vicinty of the castle for some reason 3. a) Recall that a planet's 'surface gravity' is the acceleration of an object in free-fall at the surface of that planet. For Earth, we call this g; for any other planet we label it gp. Re-derive the expression for gp based on the planet's mass and radius, Mp and Rp respectively. Hint: set the Universal Gravitational force at the planet's surface equal to the net force on an object, and then apply Newton's 2nd Law b) Assume you've measured the radius of Earth by simple geometry to be 6400 km, and you happen to

Can someone help me with number 2. Thank you

Answer 1

Initial velocity of cannonball is $u=125\,mi/hr=183.3\,ft/s$

Princess is standing at a height $h=75\,ft$

Angle of projection of cannonball $\theta=30\degree$

Using $s=ut+\frac{1}{2}at^2$

The ball reaches $h=75\,ft$ ( vertical distance ) in time $t$ , acceleration in vertical direction $a=-g$ .

Initial vertical velocity $u_y=u\sin\theta$

$h=u\sin\theta t-\frac{1}{2}gt^2$

$75=183.3\,\sin30\degree t-\frac{1}{2}*9.81\,t^2$

$-4.905\,t^2+91.65\,t-75=0$

solving above quadratic equation, $t=0.858\,s$ or $t=17.83\,s$

The ball reaches the height $h=75\,ft$ at time $t=0.858\,s$ while going up, and at time $t=17.83\,s$ while going down.

The maximum possible distance the cannon can be placed from the castle is possible with  time $t=17.83\,s$ ,

and is  $x=u\cos\theta \,t=183.3\cos30\degree*17.83=2830.4\,ft$