Pile-driver or hammer impact

1 \includegraphics[max width=\textwidth, alt={}, center]{cb2cec83-6f8d-4c13-90a1-03bbf4e4452f-03_471_613_254_766} A metal post is driven vertically into the ground by dropping a heavy object onto it from above. The mass of the object is 120 kg and the mass of the post is 40 kg (see diagram). The object hits the post with speed \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and remains in contact with it after the impact.

1. Calculate the speed with which the combined post and object moves immediately after the impact.
2. There is a constant force resisting the motion of magnitude 4800 N . Calculate the distance the post is driven into the ground.

2 \includegraphics[max width=\textwidth, alt={}, center]{99f20949-471d-4da3-a680-ec24abf6baa5-03_510_604_260_769} A machine for driving a nail into a block of wood causes a hammerhead to drop vertically onto the top of a nail. The mass of the hammerhead is 1.2 kg and the mass of the nail is 0.004 kg (see diagram). The hammerhead hits the nail with speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and remains in contact with the nail after the impact. The combined hammerhead and nail move immediately after the impact with speed \(40 \mathrm {~ms} ^ { - 1 }\).

1. Calculate \(v\), giving your answer as an exact fraction.
2. The nail is driven 4 cm into the wood. Find the constant force resisting the motion.

1. A hammer is used to hit a tent peg into soft ground.

The hammer has mass 1.8 kg and the tent peg has mass 0.2 kg .
The hammer and tent peg are both modelled as particles and the impact is modelled as a direct collision. Immediately before the impact, the tent peg is stationary and the hammer is moving vertically downwards with speed \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) Immediately after the impact, the hammer and tent peg move together, vertically downwards, with the same speed \(v \mathrm {~ms} ^ { - 1 }\)

1. Find the value of \(v\)
2. Find the magnitude of the impulse exerted on the tent peg by the hammer, stating the units of your answer. The ground exerts a constant vertical resistive force of magnitude \(R\) newtons, bringing the hammer and tent peg to rest after they travel a distance of 12 cm .
3. Find the value of \(R\).

5. A sledgehammer of mass 12 kg is being used to drive a wooden post of mass 4 kg into the ground. A labourer moves the sledgehammer from rest at a point 0.5 m vertically above the post with constant acceleration \(16 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) directed towards the post.

1. Find the velocity with which the sledgehammer hits the post. When the sledgehammer hits the post, they both move together with common speed, \(V\).
2. Show that \(V = 3 \mathrm {~ms} ^ { - 1 }\). As the sledgehammer hits the post, the labourer relaxes his grip and applies no further force. The sledgehammer and post are brought to rest by the action of a resistive force from the ground of magnitude 1500 N .
3. Find, in centimetres, the total distance that the sledgehammer and the post travel together before coming to rest.

A block \(A\) of mass 9 kg is released from rest from a point \(P\) which is a height \(h\) metres above horizontal soft ground. The block falls and strikes another block \(B\) of mass 1.5 kg which is on the ground vertically below \(P\). The speed of \(A\) immediately before it strikes \(B\) is 7 m s\(^{-1}\). The blocks are modelled as particles.

1. Find the value of \(h\). [2] Immediately after the impact the blocks move downwards together with the same speed and both come to rest after sinking a vertical distance of 12 cm into the ground. Assuming that the resistance offered by the ground has constant magnitude \(R\) newtons,
2. find the value of \(R\). [8]

A post is driven into the ground by means of a blow from a pile-driver. The pile-driver falls from rest from a height of \(1.6\) m above the top of the post.

1. Show that the speed of the pile-driver just before it hits the post is \(5.6\) m s\(^{-1}\). [2]

The post has mass \(6\) kg and the pile-driver has mass \(78\) kg. When the pile-driver hits the top of the post, it is assumed that the there is no rebound and that both then move together with the same speed.

1. Find the speed of the pile-driver and the post immediately after the pile-driver has hit the post. [3]

The post is brought to rest by the action of a resistive force from the ground acting for \(0.06\) s. By modelling this force as constant throughout this time,

1. find the magnitude of the resistive force, [4]
2. find, to 2 significant figures, the distance travelled by the post and the pile-driver before they come to rest. [4]

A tent peg is driven into soft ground by a blow from a hammer. The tent peg has mass 0.2 kg and the hammer has mass 3 kg. The hammer strikes the peg vertically. Immediately before the impact, the speed of the hammer is \(16 \text{ m s}^{-1}\). It is assumed that, immediately after the impact, the hammer and the peg move together vertically downwards.

1. Find the common speed of the peg and the hammer immediately after the impact. [3]

Until the peg and hammer come to rest, the resistance exerted by the ground is assumed to be constant and of magnitude \(R\) newtons. The hammer and peg are brought to rest 0.05 s after the impact.

1. Find, to 3 significant figures, the value of \(R\). [5]