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
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 crashing
rockets accelerating
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.
$$F \Delta t = mv - mu$$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
$$F \Delta t = m\Delta v$$ $$\frac{F \Delta t}{\Delta v} = m$$ $$\frac{(-100)(0.1)}{14-15} = m$$ $$\frac{(-100)(0.1)}{-1} = m$$ $$10 \, \mathrm{kg} = m$$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.
$$F \Delta t = m\Delta v$$ $$\Delta v = \frac{F}{m} \Delta t$$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.