Question

A stone is thrown upward from the top of a building at an angle of 30° to the horizontal and with an initial speed of 20 m/s. The point of release is 45 m above the ground. How long does it take for the stone to hit the ground? What is the stone's speed?

What is the horizontal range of the stone? Neglect air resistance.

Answer 1

The above diagram explains the motion of the stone. `A` is the point of projection,`B` is the point where the stone returns to the height of point of projection during falling down and `D` is the point where it lands. Clearly `R` is the desired range of its motion.

As,the stone is thrown at an angle of 30 w.r.t the horizontal,its vertical component of velocity becomes`u sin 30` and horizontal component is `u cos 30` (where, `u=20 m/s`)

so,time taken by the stone to reach point `B` from point `A` will be `(2u sin 30)/g` (using `v=u-at` we get time to reach its highest point,then multiply with `2` as it will take the same time to come back)

so,time taken to reach `B` from `A` becomes `2 s`

Now,at point `B`,the stone will have a downward velocity of `u sin 30` and a horizontal velocity of `u cos 30`

so,the velocity with which it will vertically hit the ground can be calculated using `v^2=u^2=2gs`

here,`u=u sin 30` and `s= 45 m` i.e the height of the building.

so, `v^2 =1000` hence, `v=31.62 m/s`
but during hitting the ground it also has a horizontal component of velocity,hence total velocity with which it hits the ground is`sqrt` `(v^2+(u cos 30)^2)` i.e `36.05 m/s`

Now,while falling downwards from point `B` to `C` total time required will be `t=(v-usin 30)/g` or, `2.16 s`,

therefore,total time after which it will reach the ground will be `(2+2.16) s or 4.16 s`

Now, in this `2.16 s` if the stone goes by a distance of `x`(`CD` in the diagram)horizontally we can say `x= (ucos 30 *2.16)` or,`37.41 m`

and we can also calculate the length of `AB` using the same formula and it will be `AB = (u cos 30 *2)` or, `34.64 m`

so,range of the motion of the stone becomes `(34.64+37.41)` m or `72.05 m`