momentum

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.

m

$$p = mv$$

\(p\) = momentum [kg m/s] vector
\(m\) = mass [kg]
\(v\) = velocity [m/s] vector
Example: What is the momentum of a bowling ball? (in kg m/s)
Google a typical mass and velocity.
solution $$ 18 \left( \mathrm{ \frac{\color{red}{mile}}{\color{Teal}{hour}}} \right)\left(\frac{1609\,\mathrm{m}}{1 \,\color{red}{\mathrm{mile} }}\right)\left(\frac{1\, \color{Teal}{ \mathrm{hour} }}{3600\,\mathrm{s}}\right) = 8.0 \mathrm{\tfrac{m}{s}} $$
$$p=mv$$ $$p=(5.0 \, \mathrm{kg})(8.0 \, \mathrm{\tfrac{m}{s}})$$ $$p=40 \,\mathrm{kg\tfrac{m}{s}}$$

Momentum and Force

Newton's 2nd law was originally written in terms of momentum, not mass and acceleration. We can rearrange F=ma a bit to see how they are equivalent.

$$a = \color{red}\frac{\Delta v}{\Delta t}$$
$$F = ma$$ $$F = m \color{red}\frac{\Delta v}{\Delta t}$$ $$F \Delta t = m \Delta v$$
$$F \Delta t = m(v_f - v_i)$$ $$F \Delta t = mv_f - mv_i$$ $$F \Delta t = p_f - p_i$$ $$F \Delta t = \Delta p$$

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 speeding up
cars crashing
rockets accelerating
kicking a soccer ball
punching
jumping
derivation of impulse $$a = \color{red}\frac{\Delta v}{\Delta t}$$
$$F = ma$$ $$F = m \color{red}\frac{\Delta v}{\Delta t}$$ $$F \Delta t = m \Delta v$$
$$F \Delta t = m(v_f - v_i)$$ $$F \Delta t = mv_f - mv_i$$ $$F \Delta t = p_f - p_i$$ $$F \Delta t = \Delta p$$

$$J = \Delta p$$ $$F \Delta t = \Delta p $$

\(J\) = 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.

$$F \Delta t = mv - mu$$
Example: Which will produce a greater change in velocity: doubling the force or doubling the time the force is applied?
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.

$$\Delta v = \frac{\color{blue}F \Delta t}{m} $$
Example: A medicine ball is hurtling towards you at 15 m/s. You weakly try to stop it by applying a 100 N force for 0.1 s. You don't stop the ball, but it slows to 14 m/s. Calculate the mass of the ball.
solution $$F \Delta t = m\Delta v$$ $$\frac{F \Delta t}{\Delta v} = m$$ $$\frac{(-100)(0.1)}{14-15} = m$$ $$\frac{(-100)(0.1)}{-1} = m$$ $$10 \, \mathrm{kg} = m$$
Example: Electric cars are capable of very high accelerations because they don't have to shift gears. The Tesla Model S car has a mass of 2,250 kg. It has one of the fastest 0 to 60 miles/hour accelerations at 2.8 seconds. Convert the miles/hour into m/s, and find the force produced by the car.
solution $$ 60 \left( \mathrm{ \frac{\color{red}{mile}}{\color{blue}{hour}}} \right)\left(\frac{1609\,\mathrm{ m}}{1 \,\color{red}{\mathrm{mile} }}\right)\left(\frac{1\, \color{blue}{ \mathrm{hour} }}{3600\, \mathrm{s} }\right) = 26.8 \mathrm{\tfrac{m}{s}} $$
$$F \Delta t = m\Delta v$$ $$F = \frac{m\Delta v}{\Delta t}$$ $$F = \frac{(2250)(26.8)}{2.8}$$ $$F = \frac{60300}{2.8}$$ $$F = 21536 \, \mathrm{N}$$

How does the 0 to 60 miles/hour acceleration compare to the acceleration of gravity?
solution $$F = 21536 N$$ $$F = ma$$ $$\frac{F}{m}=a$$ $$\frac{21536}{2250}=a$$ $$a = 9.57 \,\mathrm{ \tfrac{m}{s^2} }$$

It's just a bit under 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.

$$F \Delta t = m\Delta v$$ $$\Delta v = \frac{F}{m} \Delta t$$
force = N
mass = kg
Example: The slope of this graph is acceleration, and a high acceleration can be dangerous. Use the graph to identify what types of vehicles will be safer in collisions?
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.