A block ofice is sliding down the sloping roof of a house and the angle of inclination of the roof with the horizontal is 30^@ . The maximum and minimum heights of the roof from the ground are 8.1 m and 5.6 m. How far from the starting point, measured horizontally , does the block land ?[ignore friction]

A block ofice is sliding down the sloping roof of a house and the angl

1.	A block ofice is sliding down the sloping roof of a house and the angle of inclination of the roof with the horizontal is 30^@ . The maximum and minimum heights of the roof from the ground are 8.1 m and 5.6 m. How far from the starting point, measured horizontally , does the block land ?[ignore friction]
Answer» Solution :LET the highest point of the roofbe A and the lowest point be P as shown in Fig.2.69. `therefore AC=8.1m, PD =5.6m` `therefore AB=AC-BC=AC-PD=8.1-5.6=2.5m` `AP=(AB)/(sin 30^@)=(2.5)/(1/2)=5m` Let the velocity of the block at P be v . Considering the motion of the block from A to P, `v^2=2g sin 30^2xxAP =2xx9.8xx1/2xx5 =49 or, v=7ms^(-1)`. The horizontal and VERTICAL components of the velocity at P are `v cos 30^@ =(7sqrt(3))/(2) ms^(-1) and v sin 30^@ =7/2 m*s^(-1)` respectively. Let the total taken by the block tocome from P to E be t. Considering the vertical motion of the block , `5.6=7/2t+1/2xx9.8xxt^2 or, 7t^2 +5t-8=0` `therefore t=(-5pmsqrt(25+224))/(14)=0.77s`[takingthe POSITIVE value of t] Now, `DE=v cos30^@ xxt=(7sqrt(3))/(2)xx0.77=2.7sqrt(3) m` `therefore CE=CD+DE=2.5sqrt(3)+2.7sqrt(3)` [`tan 30^@ =(AB)/(PB) or, 2.5 SQRT(3)m and , DC =PB =2.5 sqrt(3)m]` =`5.2 sqrt(3)=9m`.