Equations of Motion

These equations give us the average velocity and acceleration

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

The Equations of Motion

If an object's acceleration is constant the equations of motion are also true. They give exact values of intial and final velocity, not averages. Here is the derivation of these equations.

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

\(\Delta x\) = distance [m] vector

\(\Delta t\) = time period [s] vector

\(v\) = final velocity [m/s] vector

\(u\) = initial velocity [m/s] vector

\(a\) = acceleration [m/s²] (constant)vector

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 a bullet would stop quicker because of air friction.)

list known and unknown variables

$$u = 400 \tfrac{m}{s}$$ $$a = -9.8 \tfrac{m}{s^{2}}$$ $$\Delta t = 40.8s$$ $$\Delta x = ?$$

plug variables into matching equation

$$\Delta x = u\Delta t + \tfrac{1}{2} a \Delta t^{2}$$ $$\Delta x = 400 (40.8) + \tfrac{1}{2} (-9.8)(40.8^{2})$$ $$\Delta x = 16320 - 8157$$ $$\Delta x = 8163m$$

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?

list known and unknown variables

$$u = 30 \tfrac{m}{s}$$ $$v = 0\tfrac{m}{s}$$ $$\Delta x = 10m$$ $$a = ?$$

isolate the unknown variable

$$v^{2} = u^{2}+2a \Delta x$$ $$v^{2} - u^{2}=2a \Delta x$$ $$\frac{v^{2} - u^{2}}{2 \Delta x}=a$$

plug in variables

$$\frac{0^{2} - (30^{2})}{2 (10)}=a$$ $$\frac{-900}{20}=a$$ $$-45 \tfrac{m}{s^{2}}=a$$
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?
(1m = 3.3ft , 1 mile = 1609m)

convert variables to correct units

$$ 170\left(\frac{\color{red}{mile}}{\color{blue}{hour}}\right)\left(\frac{1609 m}{1 \color{red}{mile}}\right)\left(\frac{1 \color{blue}{hour}}{3600s}\right) = 76\tfrac{m}{s} $$ $$u = rest = 0$$ $$\Delta x = 6000ft\scriptsize \left(\frac{1m}{3.3ft}\right)\normalsize = 1818m$$ $$a = ?$$

plug variables into matching equation

$$v^{2} = u^{2}+2a \Delta x$$ $$v^{2} - u^{2}=2a \Delta x$$ $$\frac{v^{2} - u^{2}}{2 \Delta x}=a$$ $$\frac{76^{2} - (0^{2})}{2 (1818)}=a$$ $$\frac{5776}{3636}=a$$ $$1.589 \tfrac{m}{s^{2}}=a$$

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 free fall?
$$u = rest = 0$$ $$v = -6.0\small\frac{m}{s} $$ $$a = -9.8 \small\frac{m}{s^{2}} $$ $$\Delta t = ? $$
$$v = u+a \Delta t$$ $$\large\frac{v - u}{a}\normalsize= \Delta t $$ $$\large\frac{-6 - 0}{-9.8}\normalsize= \Delta t $$ $$0.61s = \Delta t$$
Example: A 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 $$\Delta x = -4000m$$ $$u = rest = 0$$ $$a = lookup = -1.315\tfrac{m}{s^{2}}$$ $$v = ?$$ $$\Delta t = ?$$

solving for v

$$v^{2} = u^{2}+2a \Delta x$$ $$v^{2} = 0^{2}+2(-1.315)(-4000)$$ $$v^{2} = 10520 $$ $$v = \pm102.6\frac{m}{s} $$

solving for Δt

$$v = u+a \Delta t$$ $$\large \frac{v - u}{a} \normalsize = \Delta t$$ $$\large \frac{-102.6 - 0}{-1.315} \normalsize = \Delta t $$ $$78.0s = \Delta t $$