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Linear momentum and impulse

Publisher Summary

This chapter discusses linear momentum and impulse. The momentum of a body is defined as the product of its mass and its velocity, that is, momentum = mu, where m = mass (in kg) and u = velocity (m/s). The unit of momentum is kg m/s. Momentum, being a vector quantity, has both magnitude and direction. Forces that act for very short periods of time are called impulsive forces. The product of the impulsive force and the time during which it acts is called the impulse of the force and is equal to the change of momentum produced by the impulsive force. Examples where impulsive forces occur include when a gun recoils and when a free-falling mass hits the ground. However, in practice when impulsive forces occur, momentum is not entirely conserved and some energy is changed into heat, noise, and so on.

(i) The momentum of a body is defined as the product of its mass and its velocity,

$i .e . momentum =mu$ $i .e . momentum =mu$

where

$m = mass (in kg)$ $m = mass (in kg)$

and

$u = velocity (in m/s) .$ $u = velocity (in m/s) .$

The unit of momentum is kg m/s.

(ii) Since velocity is a vector quantity, momentum is a vector quantity, i.e. it has both magnitude and direction.

(i) Newton’s first law of motion states:

‘a body continues in a state of rest or in a state of uniform motion in a straight line unless acted on by some external force.’

Hence the momentum of a body remains the same provided no external forces act on it.

(ii) The principle of conservation of momentum for a closed system (i.e. one on which no external forces act) may be stated as:

‘the total linear momentum of a system in any given direction is a constant’.

(iii) The total momentum of a system before collision in a given direction is equal to the total momentum of the system after collision in the same direction. In Figure 26.1, masses m₁ and m₂ are travelling in the same direction with velocity u₁ > u₂. A collision will occur, and applying the principle of conservation of momentum:

total momentum before impact = total momentum after impact

$i . e . m_{1} u_{1} + m_{2} u_{2} = m_{1} v_{1} + m_{2} v_{2}$ $i . e . m_{1} u_{1} + m_{2} u_{2} = m_{1} v_{1} + m_{2} v_{2}$

where v₁ and v₂ are the velocities of m₁ and m₂ after impact.

(i) Newton’s second law of motion may be stated as:

‘the rate of change of momentum is directly proportional to the applied force producing the change, and takes place in the direction of this force’.

In the SI system, the units are such that:

$\begin{array}{l} the applied force = rate of change of momentum \\ = \frac{change of momentum}{change of time} \end{array}$ $\begin{array}{l} the applied force = rate of change of momentum \\ = \frac{change of momentum}{change of time} \end{array}$ (1)

(1)

(ii) When a force is suddenly applied to a body due to either a collision with another body or being hit by an object such as a hammer, the time taken in equation (1) is very small and difficult to measure. In such cases, the total effect of the force is measured by the change of momentum it produces.

(iii) Forces which act for very short periods of time are called impulsive forces. The product of the impulsive force and the time during which it acts is called the impulse of the force and is equal to the change of momentum produced by the impulsive force,

i.e. impulse=applied force × time=change in linear momentum.

(iv) Examples where impulsive forces occur, include when a gun recoils and when a free-falling mass hits the ground. Solving problems associated with such occurrences often requires the use of the equation of motion:

$v^{2} = μ^{2} + 2 a s .$ $v^{2} = μ^{2} + 2 a s .$

4. When a pile is being hammered into the ground, the ground resists the movement of the pile and this resistance is called a resistive force.

Newton’s third law of motion may be stated as:

‘for every force there is an equal and opposite force’.

The force applied to the pile is the resistive force. The pile exerts an equal and opposite force on the ground.

5. In practice when impulsive forces occur, momentum is not entirely conserved and some energy is changed into heat, noise, and so on.

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Table of Contents for Chapter 26: Linear momentum and impulse

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Publisher Summary

Table of Contents for
Chapter 26: Linear momentum and impulse