To Calculate average velocity and average acceleration we can always use these equations:

$$v_{avg} = \frac{\Delta x}{\Delta t} \ \ \ \ \ a_{avg} = \frac{\Delta v}{\Delta t} $$

If an object's acceleration is constant the equations of motion are also true. Notice that they give exact values, not averages. Here is the derivation of these equations.

The Equations of Motion

$$v_{f} = v_{i}+a \Delta t$$ $$\Delta x = u\Delta t + \small\frac{1}{2}a \Delta t^{2}$$ $$\Delta x = \small\frac{1}{2}\normalsize(v+u)\Delta t$$ $$v^{2} = u^{2}+2a \Delta x$$ $$\Delta x = v\Delta t - \small\frac{1}{2}a \Delta t^{2}$$

Δx = distance [m] vector

Δt = time [s]

v = final velocity [m/s] vector

u = initial velocity [m/s] vector

a = acceleration [m/s²] vector

To choose which equation to use write down the one unknown and all the known variables. Match your variables to one of the equations above.

Example: A bullet aimed straight up leaves the barrel of a gun at 400m/s. It accelerates down at -9.8m/s². If the bullet travels for 40.8s before stopping how far up did it go?
(In the real world the bullet would stop quicker because of air friction.)

solution

variables	algebra
$ u = 400 \small\frac{m}{s}$	$\Delta x = u\Delta t + \small\frac{1}{2}\normalsize a \Delta t^{2}$
$ a = -9.8 \small\frac{m}{s^{2}} $	$\Delta x = 400 (40.8) + \small\frac{1}{2}\normalsize (-9.8)(40.8^{2})$
$\Delta t = 40.8s$	$\Delta x = 16320 - 8157$
$\Delta x = ?$	$\Delta x = 8163m$

Example: A car moving at 30m/s puts on its brakes and comes to a stop in 10m. What is the acceleration of the car?

solution

variables	algebra
$ u = 30 \small\frac{m}{s}$	$v^{2} = u^{2}+2a \Delta x$
$ a = ? $	$v^{2} - u^{2}=2a \Delta x$
$v = 0$	$\large\frac{v^{2} - u^{2}}{2 \Delta x}\normalsize=a$
$\Delta x = 10m$	$\large\frac{0^{2} - (30^{2})}{2 (10)}\normalsize=a$
	$\large\frac{-900}{20}\normalsize=a$
	$-45\small\frac{m}{s^{2}}\normalsize=a$

Example: A plane takes off at a speed of 170 miles/hour while accelerating from rest on a runway that is 6,000ft long. What is the acceleration of the plane?

solution

convert	variables	algebra
$ v = 170 (\frac{miles}{hour})(\frac{1609m}{1miles})(\frac{1hour}{3600s})$	$ a = ? $	$v^{2} = u^{2}+2a \Delta x$
$ v = 76 \small\frac{m}{s}$	$ u = rest = 0$	$v^{2} - u^{2}=2a \Delta x$
$ \Delta x = 6000ft (\frac{1m}{3.3ft})$	$v = 76 \small\frac{m}{s}$	$\large\frac{v^{2} - u^{2}}{2 \Delta x}\normalsize=a$
$ \Delta x = 1818m $	$ \Delta x = 1818m $	$\large\frac{76^{2} - (0^{2})}{2 (1818)}\normalsize=a$
		$\large\frac{5776}{3636}\normalsize=a$
		$1.589\small\frac{m}{s^{2}}\normalsize=a$

Acceleration of Gravity

On earth everything is pulled down at 9.8 m/s². Generally the letter g is used for this number. The rate is the same for cars, birds, puppies, apples, balloons, and everything. Other effects like air friction, thrust, and buoyancy can change the precived acceleration. Acceleration from gravity also changes based on the mass of the planet and your distance to the center of the planet.

g = acceleration from gravity on earth = 9.8 m/s²

What is the acceleration in m/s² on the surface of Mars? (check Wolfram alpha)

What is the acceleration in m/s² on the surface of the Moon? (check Wolfram alpha)

Example: A sleeping cat falls from rest off of a ledge. If the cat hits the ground moving at 6.0 m/s how long was the cat in freefall?

solution

variables	algebra
$u = rest = 0$	$ v = u+a \Delta t $
$ v = -6.0\small\frac{m}{s} $	$\large\frac{v - u}{a}\normalsize= \Delta t$
$ a = -9.8 \small\frac{m}{s^{2}} $	$\large\frac{-6 - 0}{-9.8}\normalsize= \Delta t$
$ \Delta t = ?$	$0.61s = \Delta t$

Example: An meteor 4,000m above Europa, a moon of Jupiter, falls from rest. What is the impact velocity of the meteor? How long does it take for the meteor to hit the surface? (look up the gravity on Europa)

solution

variables	v algebra	Δt algebra
$u = rest = 0$	$v^{2} = u^{2}+2a \Delta x$	$v = u+a \Delta t$
$\Delta x = 4,000m$	$v^{2} = 0^{2}+2(4.313)(4,000)$	$\large \frac{v - u}{a} \normalsize = \Delta t$
$v = ?$	$v^{2} = 34504$	$\large \frac{185.8 - 0}{4.313} \normalsize = \Delta t$
$\Delta t = ?$	$v = 185.8\frac{m}{s}$	$43.1s = \Delta t$
$a = 4.313\small\frac{m}{s^{2}}$

variables	algebra
\( u = 400 \small\frac{m}{s}\)	\(\Delta x = u\Delta t + \small\frac{1}{2}\normalsize a \Delta t^{2}\)
\( a = -9.8 \small\frac{m}{s^{2}} \)	\(\Delta x = 400 (40.8) + \small\frac{1}{2}\normalsize (-9.8)(40.8^{2})\)
\(\Delta t = 40.8s\)	\(\Delta x = 16320 - 8157\)
\(\Delta x = ?\)	\(\Delta x = 8163m\)

variables	algebra
\( u = 30 \small\frac{m}{s}\)	\(v^{2} = u^{2}+2a \Delta x\)
\( a = ? \)	\(v^{2} - u^{2}=2a \Delta x\)
\(v = 0\)	\(\large\frac{v^{2} - u^{2}}{2 \Delta x}\normalsize=a\)
\(\Delta x = 10m\)	\(\large\frac{0^{2} - (30^{2})}{2 (10)}\normalsize=a\)
	\(\large\frac{-900}{20}\normalsize=a\)
	\(-45\small\frac{m}{s^{2}}\normalsize=a\)

convert	variables	algebra
\( v = 170 (\frac{miles}{hour})(\frac{1609m}{1miles})(\frac{1hour}{3600s})\)	\( a = ? \)	\(v^{2} = u^{2}+2a \Delta x\)
\( v = 76 \small\frac{m}{s}\)	\( u = rest = 0\)	\(v^{2} - u^{2}=2a \Delta x\)
\( \Delta x = 6000ft (\frac{1m}{3.3ft})\)	\(v = 76 \small\frac{m}{s}\)	\(\large\frac{v^{2} - u^{2}}{2 \Delta x}\normalsize=a\)
\( \Delta x = 1818m \)	\( \Delta x = 1818m \)	\(\large\frac{76^{2} - (0^{2})}{2 (1818)}\normalsize=a\)
		\(\large\frac{5776}{3636}\normalsize=a\)
		\(1.589\small\frac{m}{s^{2}}\normalsize=a\)

variables	algebra
\(u = rest = 0\)	\( v = u+a \Delta t \)
\( v = -6.0\small\frac{m}{s} \)	\(\large\frac{v - u}{a}\normalsize= \Delta t\)
\( a = -9.8 \small\frac{m}{s^{2}} \)	\(\large\frac{-6 - 0}{-9.8}\normalsize= \Delta t\)
\( \Delta t = ?\)	\(0.61s = \Delta t\)

variables	v algebra	Δt algebra
\(u = rest = 0\)	\(v^{2} = u^{2}+2a \Delta x\)	\(v = u+a \Delta t\)
\(\Delta x = 4,000m\)	\(v^{2} = 0^{2}+2(4.313)(4,000)\)	\(\large \frac{v - u}{a} \normalsize = \Delta t\)
\(v = ?\)	\(v^{2} = 34504\)	\(\large \frac{185.8 - 0}{4.313} \normalsize = \Delta t\)
\(\Delta t = ?\)	\(v = 185.8\frac{m}{s}\)	\(43.1s = \Delta t\)
\(a = 4.313\small\frac{m}{s^{2}}\)