A bullet is fired at a block of wood suspended from the ceiling by a single rope. Assuming no consequence due to the properties of the rope itself, and given the wood is suspended 4 metres from the ceiling to its centre, the bullet hits the dead centre of the wood and causes no spin, the wood has a mass of 2kg, the bullet has a mass of 5g and embeds in the wood, no wood is lost, impact velocity was 965 m/s. What is the maximum angle the suspended block would achieve from the perpendicular?
Verified answer
Topic :-
Mechanics
Given :-
A bullet is fired at a block of wood suspended from the ceiling by a single rope. Assuming no consequence due to the properties of the rope itself and given the wood is suspended 4 metres from the ceiling to its centre, the bullet hits the dead centre of the wood and causes no spin, the wood has a mass of 2kg, the bullet has a mass of 5g and embeds in the wood, no wood is lost, impact velocity was 965 m/s.
To Find :-
What is the maximum angle the suspended block would achieve from the perpendicular ?
Solution :-
Assumptions
Mass of block = M = 2 kg
Mass of bullet = m = 5 g = 5 × 10⁻³ kg
Impact velocity = v = 965 m/s
Final velocity of (M + m) = V
Length of rope = L = 4 m
Maximum height gained by block = h
Maximum angle achieved by suspended block =
Conserving Momentum of bullet and block,
Initially block was at rest, a bullet with some velocity (impact velocity) strikes the centre of wood and gets embedded and then combined mass rises to some height forming some angle with perpendicular.
mv + M(0) = mV + MV
mv = (m + M)V
Angle of suspended block will be maximum when block will achieve its highest height.
From Newton's Laws of Motion,
V² - (0)² = 2gh
V² = 2gh
From figure,
Answer :-
The maximum angle the suspended block would achieve from the perpendicular is approximately 21.87°.
Note : For better understanding, refer to the attachment.