Projectile on rough surface

6 A small block of mass 0.15 kg moves on a horizontal surface. The coefficient of friction between the block and the surface is 0.025 .

1. Find the frictional force acting on the block.
2. Show that the deceleration of the block is \(0.25 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). The block is struck from a point \(A\) on the surface and, 4 s later, it hits a boundary board at a point \(B\). The initial speed of the block is \(5.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
3. Find the distance \(A B\). The block rebounds from the board with a speed of \(3.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and moves along the line \(B A\). Find
4. the speed with which the block passes through \(A\),
5. the total distance moved by the block, from the instant when it was struck at \(A\) until the instant when it comes to rest.

7 \includegraphics[max width=\textwidth, alt={}, center]{fa0e0e0d-b0a6-44e0-8b4f-4923e235c6c6-3_168_803_1909_671} The frictional force acting on a small block of mass 0.15 kg , while it is moving on a horizontal surface, has magnitude 0.12 N . The block is set in motion from a point \(X\) on the surface, with speed \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). It hits a vertical wall at a point \(Y\) on the surface 2 s later. The block rebounds from the wall and moves directly towards \(X\) before coming to rest at the point \(Z\) (see diagram). At the instant that the block hits the wall it loses 0.072 J of its kinetic energy. The velocity of the block, in the direction from \(X\) to \(Y\), is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at time \(t \mathrm {~s}\) after it leaves \(X\).

1. Find the values of \(v\) when the block arrives at \(Y\) and when it leaves \(Y\), and find also the value of \(t\) when the block comes to rest at \(Z\). Sketch the velocity-time graph.
2. The displacement of the block from \(X\), in the direction from \(X\) to \(Y\), is \(s \mathrm {~m}\) at time \(t \mathrm {~s}\). Sketch the displacement-time graph. Show on your graph the values of \(s\) and \(t\) when the block is at \(Y\) and when it comes to rest at \(Z\).

4 Particles \(P\) and \(Q\) are moving in a straight line on a rough horizontal plane. The frictional forces are the only horizontal forces acting on the particles.

1. Find the deceleration of each of the particles given that the coefficient of friction between \(P\) and the plane is 0.2 , and between \(Q\) and the plane is 0.25 . At a certain instant, \(P\) passes through the point \(A\) and \(Q\) passes through the point \(B\). The distance \(A B\) is 5 m . The velocities of \(P\) and \(Q\) at \(A\) and \(B\) are \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), respectively, both in the direction \(A B\).
2. Find the speeds of \(P\) and \(Q\) immediately before they collide.

A brick of mass 3 kg slides in a straight line on a horizontal floor. The brick is modelled as a particle and the floor as a rough plane. The initial speed of the brick is 8 m s\(^{-1}\). The brick is brought to rest after moving 12 m by the constant frictional force between the brick and the floor.

1. Calculate the kinetic energy lost by the brick in coming to rest, stating the units of your answer. [2]
2. Calculate the coefficient of friction between the brick and the floor. [4]

A stone that weighs 15 kg is propelled across the ice in an ice rink with an initial speed of \(4 \text{ m s}^{-1}\). The coefficient of friction between the stone and the ice is \(0.017\). How far does the stone slide before it comes to rest? [5]