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Velocity-time graphs For a vel-t graph with: Constant acceleration, a. initial velocity, vi. final velocity, vf. And a graph that looks like the one
on the board.
Velocity-time graphs From the slope-intercept equation for a
line: y = mx + b. y = velocity (v) x = time (t) m = acceleration (a) b = initial velocity (vi) Equation becomes: v = vi + at Solving for the final velocity (vf) at any time
t, this equation is rewritten as vf = vi + at.
Kinematics vf = vi + at is the first kinematic
equation. Kinematic equations describe the
motion of an object undergoing constant (unchanging) acceleration.
There are 4 kinematic equations. Let’s derive the rest of them!
Kinematics On a vel-t graph, the displacement
(distance) is found by calculating the area under the curve.
This area is made up of a triangle (A) and a rectangle (A ).
What is the area of the triangle? A = ½ (vf – vi)t What is the area of the rectangle? A = vit
Kinematics Total area = A + A = ? Distance = A + A = ½ (vf – vi)t +
vit d = ½ (vf – vi)t + vit Solve for d.
d = ½ (vi + vf)t This is the 2nd kinematic equation.
Kinematics Substitute vf=vi + at into d=½ (vi +
vf)t d=½ (vi + vi + at)t d=½ (2vi+ at)t d=vi t+ ½ at2
This is the 3rd kinematic equation
Kinematics Solve vf = vi + at for t. t = (vf - vi)/a Substitute into d=½ (vi + vf)t d=½ (vi + vf) (vf - vi)/a 2ad=vf
2 - vi2
Rewritten as vf2 = vi
2 + 2ad This is the 4th kinematic equation.
Kinematics Table to put into notes To solve equations: Write down what you know Write down what you are seeking Write down the equation that
contains those variables.
Kinematics Examples - Nimitz
Runway length is 310 ft or 94.5 m.
Initial velocity = 0 m/s
Final velocity = 170 mph or 76.0 m/s
A steam-powered catapult accelerates the planes to launch by attaching to the nose cone.
Kinematics Examples - Nimitz
What is the acceleration required for a successful launch?
You know: vi = 0 m/s, vf = 76.0 m/s, and d = 94.5 m. You are seeking a. What equation contains those variables?
vf 2 = vi
2 + 2ad (76.0 m/s)2 =(0 m/s)2 + 2a(94.5 m) a = 30.6 m/s2 (with 3 sig figs) The force felt by the pilot is about 3 g’s.
Kinematics Examples - Nimitz
What is the time during which this acceleration takes place?
You know: vi = 0 m/s, vf = 76.0 m/s, and d = 94.5 m. You are seeking t. What equation contains those variables?
d = ½ (vi + vf)t 94.5 m = ½ (0 m/s + 76.0 m/s)t t = 2.49 s (with 3 sig figs) Yes – that IS 0 to 170 mph in 2.5 seconds!
Kinematics Examples - Nimitz
Aircraft land by catching a tailhook on an arresting wire.
The landing runway is 315 ft or 96 m.
The initial velocity is 150 mph or 67 m/s.
The total time to land is 2.0 s.
Kinematics Examples - Nimitz
What was the acceleration during this time? You know: vi = 67 m/s, d = 96 m, and t = 2.0
s. You are seeking a. What equation contains those variables?
d=vi t+ ½ at2
96 m = (67 m/s)(2.0 s) + ½ a(2.0 s)2
a = -19 m/s2 (with 2 sig figs) Note the negative acceleration because the
aircraft is actually slowing down! The force felt by the pilot is about 2g’s.
Kinematics Examples A car initially moving at 17 m/s accelerates at 12
m/s2 for 180 m. During what time did this acceleration take place?
You know: vi = 17 m/s, a = 12 m/s2, and d = 180 m. You are seeking t. What equation contains those variables?
d=vi t+ ½ at2
180 m = (17 m/s) t + ½ (12 m/s2) t2
How will you solve this equation for t? Use the quadratic equation. How to do on TI calcs t = 4.2 s (with 2 sig figs)