Conservation of momentum

The Principle of the Conservation of Linear Momentum is derived from Newton’s third law of motion. That is, for every acting force there is an equal and opposite reaction force. The Principle of Conservation of Linear Momentum states that: the total momentum of as system, in a given direction, remains constant unless an external force is applied to the system in that given direction. This principle enables impact problems involving only internal forces to be easily resolved.

Consider the firing of a projectile from a gun. Equal and opposite forces are acting on the projectile and the gun will cause the projectile to move forward and the gun to move backwards. Since there are no external forces involved, if both gun and projectile had equal masses they would move with equal velocity. Fortunately, in practice, the gun has a much greater mass than the projectile so that the recoil is much less than the forward movement of the projectile.

Example

A body P of mass 15 kg moving with a velocity of 30 m/s when it collides with a second body Q of mass 5 kg which is moving in the same direction at 5 m/s. Determine the common velocity of the two bodies after impact.

By the Principle of the Conservation of Linear Momentum:

Momentum before impact =momentum after impact

Mass ×velocity before impact =mass ×velocity after impact

15 kg ×30 m/s +5kg ×5 m/s =(15 kg +5 kg)vI where v_I =velocity after impact

v_I =23.75 m/s

Figure 3.10(a) shows that had the two bodies been moving in opposite directions before impact then the velocity of the second body Q would be considered as a negative quantity and the calculation would be as follows:

Momentum before impact=momentum after impact

15 kg(+30 m/s) +5 kg(-5 m/s) =(15 kg +5 kg)v_I

v_I =21.25 m/s

 Impact of a fluid jet on a fixed body

• If the jet is deflected from its line of motion then, from Newton’s first law, a force must be acting on it.
• If the force that is acting on the jet is proportional to the rate of change of momentum and acts in the direction of the change of momentum, then Newton’s second law is applicable.
• If the force exerted by the body on the jet to change its momentum then this force must be equal and opposite to the force exerted by the jet on the body.

Since this example involves the impact of a fluid jet, then it is concerned with jet flow rate and Newton’s second law of motion can be written as:

Force =rate of change of momentum

=(mass×change of velocity)/time taken

=mass flow rate ×change of velocity in the direction of the force

 

Example

A water jet of 50 mm diameter impacts perpendicularly on a fixed body with a velocity of 75 m/s. Determine the force acting up the body.

Assume that the jet of water will spray out radially when it impacts on the body; that is, it is deflected through an angle of 90°. Therefore the jet will lose all its momentum at right angles to the body as shown in Fig 1.

Area of jet =(π/4)  × (50/1000)²m²= 6.25 π ×10^-4

Volume flow rate = 6.25 π ×10^-4m²×75 m/s =0.047 π m³/s

Mass flow rate (water) =volume flow rate × density of water

=0.047 π m3/s × 10³kg/m³ =47 π kg/s

Force acting on the body =mass flow rate ×change in velocity

Note: the change in velocity is from 75 m/s to 0 m/s immediately the jet impacts the body.

Therefore:

Force acting on the body =47 π kg/s ×75 m/s = 3525 N =3.525 kN