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Linear Momentum, Impulse

Linear momentum​

Linear momentum, also called translational momentum or just momentum, is the product of mass and velocity of an object. Because mass is a scalar quantity and velocity is a vector quantity, linear momentum, as their product, is a velocity quantity. Its SI unit is kg⋅m/skg\cdot m/s.

p⃗=mv⃗\vec{p}=m\vec{v}

Intuitively, you can think of momentum as how hard to stop a moving object. If two objects have the same speed, the object with larger mass has more momentum; if two objects have the same mass, the object with faster speed has more momentum.

Force as the rate of change in momentum​

With the definition of acceleration a⃗=lim⁡Δt→0Δv⃗Δt\vec{a}=\lim\limits_{\Delta t\to 0}\frac{\Delta \vec{v}}{\Delta t} and Newton's second law F⃗net=ma⃗\vec{F}_{net}=m\vec{a}, we can conclude

F⃗net=mlim⁡Δt→0Δv⃗Δt=lim⁡Δt→0mΔv⃗Δt=lim⁡Δt→0Δp⃗Δt\vec{F}_{net}=m\lim_{\Delta t\to 0}\frac{\Delta \vec{v}}{\Delta t}=\lim_{\Delta t\to 0}\frac{m\Delta\vec{v}}{\Delta t}=\lim_{\Delta t\to 0}\frac{\Delta \vec{p}}{\Delta t}

In other words, the rate of change in an object's momentum is equal to the net force acting on it.

  • If the velocity of an object is 0, its position stays the same.
  • If the acceleration of an object is 0, its velocity stays the same.
  • If the net force of an object is 0, its momentum stays the same.

Impulse and momentum​

In physics, impulse J⃗\vec{J} is the integral of a force over time. If the force does not change over this time, we can write

J⃗=F⃗t\vec{J}=\vec{F}t

Impulse-momentum theorem states that impulse equals to change in momentum

J⃗=Δp⃗\vec{J}=\Delta \vec{p}

For sports that hit a ball with a racket or a stick, such as tennis, golf, baseball, and badminton, it is often recommended to "follow through all the way like a pro". Because if you contact the ball with longer duration of time, it will generate more impulse, and thus more changes in momentum.

Another example is falling from a high place. You have a task to change your momentum to 0. The required change in momentum depends on your landing velocity, which is not easy to reduce in the air. During landing, based on the impulse-momentum theorem, you can either take large force over a short period of time, or take smaller force over a long period of time. Smaller force is most likely better.

How can you land longer? Try to land with large muscle groups, land on mud, dirt, water, grass, trampoline, or even a pile of hay, fruits or vegetables, and roll. That increases your contact time with the ground.