The mass invariance in classical mechanics enables us to take the mass out of the differential equation given by Newton’s second law :

This is important from the perspective that external force can be expressed in terms of acceleration :

A situation, where measurement of force is required, assumes that mass is invariant. We proceed to measure force either in terms of acceleration or change in momentum in equivalent manner.

** Force as product of mass and acceleration **

#### Example 1

Problem : A bullet of 10 gram enters into a wooden target with a speed of 200 m/s and comes to stop with constant deceleration. If linear penetration into the wood is 10 cm, then find the force on the bullet.

A bullet hits wooden target |
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Solution : The motion of the bullet is resisted by wood, which applies a force in a direction opposite to that of the bullet. We assume wood to be uniform. The resistance offered by it, therefore, is constant. Now, as force is constant, resulting deceleration is also constant. Applying equation of motion,

u = 500 m/s ; v = 0; x = 10/100 = 0.1 m.

Force on the bullet,

F = ma = - 0.01 x 200000 = -2000 N

Note that bullet, in turn, applies 2000 N of force on the wooden target. This explains why bullet hits are so fatal for animals. Further note that we measured force as product of mass and acceleration.

** Force as a time rate of change in linear momentum **

#### Example 2

Problem : A ball of mass “m” with a speed “v” hits a hard surface as shown in the figure. The ball rebounds with the same speed and at the same angle, θ, with vertical as before. Find average force acting on the ball.

A ball strikes the hard surface |
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Solution : We can determine force either (i) by measuring acceleration or (ii) by measuring change in linear momentum during the motion. Here, we take the second approach.

The linear momentum (remember that it is a vector) before hitting surface is :

Components of linear momentum before hitting surface |
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The linear momentum (remember that it is a vector) after hitting surface is :

Components of linear momentum after hitting surface |
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The change in linear during contact with surface, Δp,

The average force is :

Note that force is acting in vertical upward direction.