Download Easy to see when the projectile initial angle is 45 the range is a

Complementary angles will produce the same range.
– The maximum height will be different for the two angles.
– The times of the flight will be different for the two angles.
Easy to see when the projectile initial angle is 45 the range is a maximum,
however for range values less than the maximum there are 2 possible initial
projectile angles and consequently 2 different trajectory paths.
For example, 30 and 60 degrees projectiles find the same mark.
The horizontal velocity component is greater for the 30 degree than the 60,
however a projectile launch at 30 degrees spends less time in the air.
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Range Angle
Solve for t
Consider the case when the projectile returns to y=0
Substitute for t
found in the above x equation
where now x=R
Make the substitution
Square both sides
range
Make the following substitution Quadratic equation Using the general form for quadratic solution Example problem
What are the possible launch angles?
Clicker
The spring‐loaded gun can launch projectiles at a) 15°
different angles with the same launch speed. At what angle should the projectile be launched in order to travel the greatest distance before landing?
b) 30°
c) 45°
d) 60°
e) 75°
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Clicker Quiz
During the Apollo 14 mission astronaut Alan Shepard hit several golf shots while on the moon. Again assume an initial speed of 70m/s at angle of 30 degrees for the golf ball. The gravitation constant on the surface of the moon is ~1/6 of that on Earth. Assuming the ball lands at the same elevation, how far did it travel in the horizontal direction? gm=1.6m/s2
A) 363m B) 2.7km
C) 740m
D) 805m
E) 630m
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If we want to determine the range of the projectile from the launcher we need to know v0
So how can we determine speed of the ball
as it leaves the launcher??
We measure the height of the
projectile for a vertical launch
h= 220
Rmax= 4.4m
R (30deg) = v0= Sqrt(2*9.81*2.2)=6.6m/s
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Air Resistance is a Drag
ax = ‐k f(v) vx
ay = ‐kf(v) vy ‐ gy
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We have investigated the consequences of acceleration on trajectories without consideration of the causes of acceleration. Actually acceleration is a consequence of a more fundamental occurrence in Nature. Forces
•Forces in everyday experience
– Push or Pull on an object to move it
– Throw or kick a ball
– Push or Pull on an object and not be able to move it
•Forces are what cause any change in the velocity of an object.
– A force is that which causes an acceleration
Use the Force
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For simplicity sake, all forces (interactions) between objects can be placed into two broad categories:
Contact Forces
Action‐at‐a‐Distance Forces
Frictional Force
Gravitational Force
Tension Force
Electrical Force
Normal Force
Magnetic Force
Air Resistance Force
Applied Force
Spring Force
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Now let’s multiply this equation by mass m
Recall mass is a positive scalar quantity and does not effect direction
m
Momentum P is the product of the mass and velocity of an object. Like velocity, momentum is a vector quantity, possessing a direction as well as a magnitude. Force is defined as the product of the mass and acceleration of an object. External force on a particle can influence it’s momentum and motion. Vector
oh yeah
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Rewrite the equation The change in momentum is caused by a net external Force F over a duration of time t
If there are no forces (interactions) Newton’s First Law of Motion or Law of Inertia
A body at rest remains at rest and a body in motion continues to move at
constant velocity along a straight line unless acted upon by an external force.
Air Hockey Puck Demo
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