6.5 Change in Momentum and the consequences of Newton's Second Law of Motion

When a resultant force acts on an object for any length of time it will cause a change in momentum by changing the velocity.

The resultant force causes a change in speed or direction in the direction of the resultant force.

The resultant force causes a change in momentum in the direction of the resultant force.

A way of stating Newton's 2nd law is to say the change in momentum is proportional to the size of the resultant force and the time interval during which the force is acting on the object.

This statement is justified by following the three mathematical steps below.

 

A resultant force on an object will cause it to accelerate or decelerate.

force = mass x acceleration

F = ma   (the mathematical expression of Newton's 2nd Law of Motion)

 

NOW acceleration is the change in velocity in a specific change in time.

a = ∆v / ∆t  (the usual formula for acceleration)

a = acceleration in m/s², v in m/s,  time t in seconds s

 

If you combine these two equations you get

force = mass x change in velocity / time

F = m ∆v / ∆t = m(v-u) / ∆t  

(where v and u are the final and initial velocities)

but, for a given object of mass m, m ∆v = ∆mv = ∆p = change in momentum

so can write Newton's 2nd law equation as

F =  ∆mv / ∆t = ∆p / ∆t   (force is equal to the rate of change of momentum)

 

These equations express force as the rate of change of momentum

and ∆p = ∆mv = F x ∆t  

(remember ∆t is the time over which the resultant force acts)

F x ∆t is sometimes called the impulse.

 

This means that a force applied to an object over any period of time must change that object's velocity.

From the point of view of solving calculation problems you need to be pretty familiar with all this maths!

 

In calculations you can then use Newton’s 2nd law of motion (F = ma) as follows:

force (newton, N) = change in momentum (kilogram metre per second, kg m/s) / time (second, s)

F = ∆mv / ∆t = (mv - mu) / ∆t,   where = mass of object kg, u = initial velocity m/s, v = final velocity m/s

this means the rate of change of momentum is a directly related to the resultant force or force applied,

so you can say the force equals the rate of change of momentum,

the equation can be written as    F = mΔv / Δt    since m is constant for a given object

which can be expressed as ...

change of momentum (kg m/s) = resultant force (N) × time for which it acts (s)

∆p = ∆mv = F x ∆t  

(the product of the resultant force x time is sometimes called the impulse)

(if the force is varying, then the average force is used in the calculation, but I don't think this is needed for GCSE)

Consequences of the force equalling the rate of change of momentum

For any moving object the faster the change in momentum (Δmv) occurs, the greater the force (F) involved, and the shorter the time taken (∆t).

You should be able to see this from the equation ∆mv = F x ∆t

for a given mv, increase in F means a decrease in ∆t,

for a given mv, decrease in F means an increase in ∆t,

 

This has serious implications for e.g. car crashes.

The faster the crash happens the bigger the force on the car and its occupants.

This sudden change momentum change in such a short time results in a great impact force.

The smaller Δt for a given momentum change, the greater the force involved and the greater the chance of injury.

If you can make the rapid deceleration occur over a longer time (increasing Δt) you reduce the resultant force and decrease the chance of serious injury.

That is what crumple zones are for in the design of modern cars.

In terms of F = ma, you are trying to reduce a and so decrease F.

For more details on this issue see Forces and motion Section 5.

5. Stopping distances, impact forces - example calculations

Note on impulse - a vector quantity 

(you may need to appreciate in some questions)

impulse = force x time

(time x the force is applied in the direction of action in units of Ns)

Since force = mass x acceleration (F = ma),

and acceleration a = Δv/Δt, then ...

impulse = mass x acceleration x time, and so ...

impulse = mass x change in momentum = mΔv