Question

A boy standing on the top of a 40m high building projects a stone vertically upwards with an initial velocity of 10m/s. The stone eventually falls to the ground.(a)After how long will the stone strike the ground?(b)After how long will the stone pass through the point from where it was projected? (c) With what velocity will it strike the ground?

Answer 1

Initially the height of the boy from the ground is 40 m. The stone is projected vertically upwards and it reaches some height and at the point of maximum height its velocity will be zero. Now the stone comes down and it hits the ground. We can divide the trajectory into several parts and then can solve them individually.

Complete step by step solution:
For first leg of the journey:
Initial velocity, u= 10 m/s
Final velocity, v= 0
Acceleration, a=g= -10 $m/{{s}^{2}}$
Using the equation of motion to find the maximum height and time taken to reach the height,

& v=u+at \\\ & \Rightarrow 0=10-10t \\\ & \Rightarrow t=1s \\\ \end{aligned}$$ Also, finding the height attained, $$\begin{aligned} & \Rightarrow {{v}^{2}}-{{u}^{2}}=2as \\\ & \Rightarrow 0-100=-20s \\\ & \Rightarrow s=5m \\\ \end{aligned}$$ The stone has reached 5 m and its total height from the ground is 45m. Now we need to find the time taken by the stone to cover 5 m downwards so that it is at the point from where it was projected. Initial velocity, u= 0 Distance, s= 5 m Acceleration= g= 10$$m/{{s}^{2}}$$ $$\begin{aligned} & s=ut+\dfrac{a{{t}^{2}}}{2} \\\ & \Rightarrow 5=0+5{{t}^{2}} \\\ & \Rightarrow t=1s \\\ \end{aligned}$$ So, the body takes 1 s Now again considering the motion from the topmost point, the total distance to be covered is 45 m. Initial velocity, u= 0 Distance, s= 45 m Acceleration= g= 10$$m/{{s}^{2}}$$ $$\begin{aligned} & \Rightarrow {{v}^{2}}-{{u}^{2}}=2as \\\ & \Rightarrow {{v}^{2}}-0=2\times 10\times 45 \\\ & \Rightarrow {{v}^{2}}=900 \\\ & \Rightarrow v=30m/s \\\ \end{aligned}$$ Finding out the time taken to cover this distance, $$\begin{aligned} & v=u+at \\\ & \Rightarrow 30=0+10t \\\ & \Rightarrow t=3s \\\ \end{aligned}$$ So, the answers are: (a)-1+3= 4 s (b)- 1s (c)- 30 m/s **Note:** No external force is acting on the system and the only force which is driving the motion of the body towards the earth is the force of attraction due to gravity. The body is moving downwards with the acceleration due to gravity and so the value of g is positive. If the case was just the opposite, that is the body was moving upwards then we would have taken the value of g to be negative.