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Laws of Motion

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Question : 11 of 40
Marks: +1, -0
A truck starts from rest and accelerates uniformly at 2.0 m s2^{-2}. At t = 10 s, a stone is dropped by a person standing on the top of the truck (6 m high from the ground). What are (a) velocity, and (b) acceleration of the stone at t = 11 s? (Neglect air resistance.)
Solution:  
Initial velocity,u=0,a=2.0m s2,t=10 su = 0, a = 2.0 \mathrm{m\ s^{-2}} , t = 10\ \mathrm{s}
Let vv be the velocity of truck when the stone is dropped from it after t = 10 s.
∴ Using the relation, v = u + at, we get
v=0+2.0×10=20 m s1v=0+2.0\times 10=20\ \mathrm{m}\ \mathrm{s}^{-1}
Horizontal velocity of the stone when it is dropped from the truck is vx=v=20m s1.v_x = v = 20 \mathrm{m\ s^{-1}}.
As air resistance is neglected, so vxv_x = constant.
Motion in the vertical direction :
Initial velocity of the stone, vy=0v_y = 0 at t=10st = 10\,\mathrm{s}
acceleration, ay=g=10 m s2a_y = g = 10\ \mathrm{m\ s^{-2}}, time t=1110=1t = 11 - 10 = 1 s
If vyv_y be velocity of the stone after 1 s of drop (i.e. at t = 11 s,) then
vy=uy+aytv_{y}=u_{y}+a_{y} t
=0+10×1=10 m s1=0+10\times 1=10\ \mathrm{m}\ \mathrm{s}^{-1}
If v be the velocity of the stone after 11 s, then
v=vx2+vy2v=\sqrt{v_{x}^{2}+v_{y}^{2}}
=(20)2+(10)2=\sqrt{(20)^{2}+(10)^{2}}
=500=22.4 m s1=\sqrt{500}=22.4\ \mathrm{m}\ \mathrm{s}^{-1}
Let q be the angle which the resultant velocity OC of the stone makes with the horizontal direction OA i.e. with vxv_x. Then from ΔOAC,
tanθ=ACOA=vyvx=1020=0.5\tan \theta = \frac{AC}{OA} = \frac{v_y}{v_x} = \frac{10}{20}=0.5
θ=26.6\therefore \theta = 26.6^{\circ}
(b) At the moment, the stone is dropped from the truck, the horizontal force on the stone is zero,
so, ax=0a_x = 0 and ay=a_y = acceleration along vertical direction =+g=10 m s2= +g = 10\ \mathrm{m\ s^{-2}} which acts in downward direction.
∴ If a = resultant acceleration of the stone, then
a=ax2+ay2a=\sqrt{a_{x}^{2}+a_{y}^{2}}
=02+(10)2=\sqrt{0^{2}+(10)^{2}} or a=10 ms2a=10\ \mathrm{ms^{-2}}
and it acts vertically downward.
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