JamieandMaya

=Equations of Motion=

Recall:
math s(t)= $position function$ math math v(t)= $velocity function$ math math a(t)= $acceleration function$ math math v(t)=\frac{ds}{dt} math math a(t)=\frac{dv}{dt} math

Example 1: An object at rest
math \frac{ds}{dt}=0 math
 * differential equation for position function

Example 2: An object with constant velocity
math v(t)=15 math math \frac{ds}{dt}=15 math

Bicyclist riding 15 mph
math s(t)=15t+C math

Find a position function for a falling object.
math a(t)=-9.8 math math \frac{dv}{dt}=-9.8 math math v(t)= -9.8+V_0 math math s(t)= -\frac{9.8}{2}t^2+V_0 t+S_0 math

1. Find the height after t seconds.
Find the acceleration, velocity, and position equations. math a(t)=-32=\frac{dv}{dt} math math v(t)=-32t+V_0 math math v(0)=48 math math math math s(t)=-\frac{32}{2}t^2+48t+S_0 math math s(0)=432 math math s(0)=-16(0)^2+48(0)+432 math math math
 * v(t)= -32t+48**= \frac{ds}{dt}
 * s(t)=-16t^2+48t+432**

2.When does it reach the maximum height?
Maximize s(t), find when v(t)=0 math -32t+48=0 math math t= \frac{48}{32}=\frac{3}{2} math What is the max height? math s(\frac{3}{2}) math math -36+72+432=**468 feet** math

3. When does it hit the ground?
The ball will hit the ground when s(t)=0 math -16t^2+48t+432=0 math math t=**6.9 seconds** math

4. What is its impact velocity?
Plug in the time it takes to hit the ground into the velocity equation. math v(6.9)=-32(6.9)+48 math math v(6.9)= **-172.8 \frac{m}{s}** math

1. Find height after t-seconds.
math a(t)=-9.8 math math v(t)=-9.8t+V_0 math math V_0=10 math math math math s(t)=-\frac{9.8}{2}t^2+10t+S_0 math math S_0=2 math math math
 * v(t)=-9.8t+10**
 * s(t)=-4.6t^2+10t+2**

2. When does it reach the maximum height?
math v(t)=0 math math -9.8(t)+10=0 math math math
 * t=\frac{10}{9.8}**

3. What is the maximum height?
math s(\frac{10}{9.8}) math math s(\frac{10}{9.8})= -4.6(\frac{10}{9.8})^2+10(\frac{10}{9.8})+2 math math s(\frac{10}{9.8})=**13.25 meters** math

4. When does it hit the ground?
math s(t)=0 math math -4.6t^2+10t+2=0 math math math
 * t=2.2 sec**

5. What is it's impact velocity?
math v(2.2)=-9.8(2.2)+10 math math math
 * v(2.2)=-11.79 \frac{m}{s}**

=
Two balls are thrown upward from the edge of a 432 ft. cliff. The first is thrown with a speed of 48 ft/s and the other is thrown one second later at a speed of 24 ft/s. Do the balls ever pass each other?=====

__Ball #1:__
math a(t)=-32 \frac{ft}{s^2} math math v(t)=-32t+48 math math s(t)=-16t^2+48t+432 math Set v(t) equal to 0 to find when the ball reaches its maximum height. math 0=-32t+48 math math t=\frac{3}{2} sec math Set s(t) equal to 0 to find when the ball hits the ground. math 0=16t^2+48t+432 math math t=6.9 sec math Determine the position of the ball at any t value, to compare its position with the other ball's. We chose 5 seconds. math s(5)=-16(5)^2+48(5)+432 math math s(5)=272 ft math

__Ball #2:__
math a(t)=-32 math math v(t)=-32t+24 math math s(t)=-16t^2+24t+432 math Set v(t) equal to 0 to find when the ball reaches maximum height. math v(t)=0 math math -32t+24=0 math math t=\frac{3}{4} sec math Set s(t) equal to 0 to find when the ball hits the ground. math s(t)=0 math math -16t^2+24t+432=0 math math t=6.00 sec math Determine the position of the ball at the same t chosen for ball #1 above. math -16(5)^2+24(5)+432=152 ft math

A stone was dropped off a cliff and hit the ground with a speed of 120 ft/s. What is the height of the cliff?
math a(t)=-32 math math v(t)=-32t math math s(t)=-16t^2+S_0 math math v(f)=120$when$s=0 math math -120=-32t math math t=3.75 sec math math s(3.75)=-16(3.75)^2+S_0 math math 0=-16(3.75)^2+S_0 math math math
 * S_0=225 ft**

**__General Differential Equations__**
math \frac{dy}{dx}= f(x) math math y(x)= F(x)+C $where$ F'(x)=f(x) math

Example:
math \frac{dy}{dx}=e^x+2 math math F(x)=e^x+2x+C math math y(x)=e^x+2x+C math math If, y(0)=3, e^0+2(0)+C=3 math math So, C=2 math math y=e^x+2x+2 math