The concept of momentum developed slowly. Aristotle (384-322 BCE) described motion as something that needed a continuing cause, which seems reasonable when you watch carts, rocks, and people eventually stop.
In 1020 CE, Ibn Sina said that a thrown object carries an impressed tendency of motion, with outside effects like air resistance gradually reducing that motion. In the early 1600s, Galileo argued that objects can keep moving without a constant push, especially when friction is reduced.
In 1687, Newton put these ideas into a mathematical theory in the Principia. Newton called momentum the quantity of motion: how much matter an object has multiplied by how fast it is moving.
This is a simulation of a "Newton's cradle". A real Newton's cradle is made of metal balls suspended by two strings. Click and drag a ball to fling it.
When two bodies collide, momentum is transferred. Momentum is the velocity of a body multiplied by its mass. A small force can quickly stop an object with low momentum, but a large or prolonged force is required to stop an object with high momentum.
$$p = mv$$
\(p\) = momentum [kg m/s] vector\(m\) = mass [kg]
\(v\) = velocity [m/s] vector
A typical mass is 5 kg and a typical and velocity is around 18 mph.
solution
$$ 18 \left( \mathrm{ \frac{\color{red}{mile}}{\color{Teal}{hour}}} \right)\left(\frac{1609\,\mathrm{m}}{1 \,\color{red}{\mathrm{mile} }}\right)\left(\frac{1\, \color{Teal}{ \mathrm{hour} }}{3600\,\mathrm{s}}\right) = 8.0 \mathrm{\tfrac{m}{s}} $$$$p=mv$$ $$p=(5.0 \, \mathrm{kg})(8.0 \, \mathrm{\tfrac{m}{s}})$$ $$p=40 \,\mathrm{kg\tfrac{m}{s}}$$
solution
The 15 kg bike moving at 6 m/s has more momentum.
Newton's Law and Momentum
Newton's second law can be written as a change in momentum with time. Force changes an object's momentum.
derivation
If the mass stays constant, the change in momentum comes from the change in velocity.
$$\sum F = \frac{\Delta p}{\Delta t}$$ $$\Delta p = m\Delta v$$ $$\sum F = \frac{m\Delta v}{\Delta t}$$ $$a = \frac{\Delta v}{\Delta t}$$ $$\sum F = ma$$So force equals mass times acceleration is a special case of the momentum version of Newton's second law.
$$\sum F = \frac{\Delta p}{\Delta t}$$
\(\sum F\) = net force [N]\(\Delta p\) = change in momentum [kg m/s]
\(\Delta t\) = time interval [s]
solution
$$\Delta p=m\Delta v$$ $$\Delta p=m(v_f-v_i)$$ $$\Delta p=(1200)(25-5)$$ $$\Delta p=24000\,\mathrm{kg\,m/s}$$ $$\sum F=\frac{\Delta p}{\Delta t}$$ $$\sum F=\frac{24000}{8.0}$$ $$\sum F=3000\,\mathrm{N}$$The car's momentum increases by 24000 kg m/s, so the average net force is 3000 N forward.
solution
Let the ball's original direction be positive. The final velocity is negative because the ball reverses direction.
$$\Delta p=m(v_f-v_i)$$ $$\Delta p=(0.145)(-30-40)$$ $$\Delta p=-10.15\,\mathrm{kg\,m/s}$$ $$\sum F=\frac{\Delta p}{\Delta t}$$ $$\sum F=\frac{-10.15}{0.010}$$ $$\sum F=-1015\,\mathrm{N}$$The negative sign means the force is opposite the ball's original motion.
Impulse
An impulse is defined as a force applied over a period of time. Applying a larger force or longer lasting force produces a larger impulse. A large impulse produces a large change in momentum.
Examples of impulse:cars crashing
rockets accelerating
punching
jumping
derivation of impulse
Newton's 2nd law (F=ma) was originally written in terms of momentum, not mass and acceleration. We can rearrange F=ma to see it in terms of momentum.
$$a = \color{green}\frac{\Delta v}{\Delta t}$$$$F = ma$$ $$F = m \color{green}\frac{\Delta v}{\Delta t}$$ $$F \Delta t = m \Delta v$$
$$F \Delta t = m(v_f - v_i)$$ $$F \Delta t = mv_f - mv_i$$ $$F \Delta t = p_f - p_i$$ $$F \Delta t = \Delta p$$
$$J = \Delta p$$ $$F \Delta t = \Delta p $$
\(J\) = impulse [Ns, kg m/s] vector\(F\) = force [N, kg m/s²] vector
\(\Delta t\) = time period [s]
\(\Delta p\) = change in momentum [kg m/s] vector
When using impulse in solving problems you might want to unpack momentum into mass and velocity.
$$F \Delta t = mv - mu$$solution
Doubling force or time applied will produce the same increase in velocity. This is because F and Δt are in the same position in the equation.
$$\Delta v = \frac{\color{blue}F \Delta t}{m} $$solution
$$F \Delta t = m\Delta v$$ $$\frac{F \Delta t}{\Delta v} = m$$ $$\frac{(-100)(0.1)}{14-15} = m$$ $$\frac{(-100)(0.1)}{-1} = m$$ $$10 \, \mathrm{kg} = m$$Electric cars are capable of high accelerations because electric motors provide instant torque without the need to shift gears.
Example: The Porsche Taycan Turbo GT (2025) has a mass of around 2250 kg. It has one of the fastest 0 to 60 miles/hour accelerations at 1.898 seconds. Convert the miles/hour into m/s, and then find the force produced by the car.solution
$$ 60 \left( \mathrm{ \frac{\color{red}{mile}}{\color{blue}{hour}}} \right)\left(\frac{1609\,\mathrm{ m}}{1 \,\color{red}{\mathrm{mile} }}\right)\left(\frac{1\, \color{blue}{ \mathrm{hour} }}{3600\, \mathrm{s} }\right) = 26.8 \mathrm{\tfrac{m}{s}} $$$$F \Delta t = m\Delta v$$ $$F = \frac{m\Delta v}{\Delta t}$$ $$F = \frac{(2250)(26.8)}{1.898}$$ $$F = \frac{60300}{1.898}$$ $$F = 31770 \, \mathrm{N}$$
How does the 0 to 60 miles/hour acceleration compare to the acceleration of gravity?
solution
$$F = 21536 N$$ $$F = ma$$ $$\frac{F}{m}=a$$ $$\frac{31770}{2250}=a$$ $$a = 14.12 \,\mathrm{ \tfrac{m}{s^2} }$$It's almost one and half times the acceleration of gravity. How would that feel?
$$ g = 9.81 \, \mathrm{\tfrac{m}{s^2}} $$We can rearrange our impulse equation to show the relationship between velocity and time.
$$F \Delta t = m\Delta v$$ $$\Delta v = \frac{F}{m} \Delta t$$mass = kg
solution
A vehicle with more mass will experience less acceleration. A large SUV is safer in a collision then a motorcycle. On the other hand, a larger vehicle is safer for you, but more dangerous for whatever you crash into.
Another way to improve safety is to reduce the force. This can be done by increasing the total time for the collision. Like hitting an airbag instead of a steering wheel. Modern cars also lengthen the time of a collision with crumple zones.