PHYSICS - CLUTCH CH 03: 2D MOTION (PROJECTILE MOTION)lightcat-files.s3.amazonaws.com/packets/admin_physics-3-clutch-ph… · MOTION IN TWO DIMENSIONS Motion in 2D (plane) in very

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PHYSICS - CLUTCH

CH 03: 2D MOTION (PROJECTILE MOTION)

MOTION IN TWO DIMENSIONS

● Motion in 2D (plane) in very similar to 1D (straight line). We can think of it as TWO sets of 1D motion:

- These “two” motions (X & Y) are _________________ & ___________________ (same ___).

- In Physics, whenever anything is 2D, we FIRST decompose it into X & Y components (1D).

● Position, Displacement, Velocity, Acceleration are VECTORS and may need to be decomposed:

POSITION DISPLACEMENT VELOCITY ACCELERATION

O

P

O

B A

v

v = _______ = __________

___ = _______ = ________

___ = _______ = ________

a

a = _______ = __________

___ = _______ = ________

___ = _______ = ________

X Y 2D X Y 2D 2D X Y 2D X Y

EXAMPLE: The diagram shows your position (in meters) in an X-Y plane. You take 10 seconds to go from A to B, while

moving with a constant velocity. You then take 5 seconds to go from B to C, while moving with a different constant velocity.

(a) Draw position vectors (x, y, r) for point A, and displacement vectors (Δx, Δy, Δr) for intervals AB and BC.

(b) Calculate the magnitude and direction of displacement and velocity vectors for intervals AB, BC, and AC.

B

A

16 C

12 B

8

4 A

0 4 8 12 16

1 x 2D 2 x 1D

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PRACTICE: MOTION IN A 2D PLANE

PRACTICE 1: An object moves up with a constant 8 m/s while moving to the left with a constant 6 m/s. Calculate the

magnitude and direction of the object’s (total) velocity.

PRACTICE 2: An object is launched with an initial velocity of 50 m/s directed 37 degrees above the horizontal. Calculate the

X and Y components of the object’s initial velocity.

PRACTICE 3: A car is initially at rest at (0, 0) meters on an X-Y plane. The car then accelerates for 4 s with a constant 6

m/s2 directed in the +x-axis. The car then accelerates for 5 s with a constant 5 m/s2 directed at 53o above the +x-axis. Find

the magnitude and direction of the car’s velocity after the two accelerations (after 8 s hint: use v = vo + at).

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PROJECTILE MOTION: INTRO & SYMMETRIC LAUNCH

● Projectile Motion occurs when an object is launched and moves freely in 2D space.

- Remember that the first step in any 2D Physics problem is to ____________________ it into X & Y components.

- When decomposing vo, vox goes with the ____ motion, voy with the ____ motion (vo no longer useful).

- So Projectile Motion is _______________ motion WITH _______________________ horizontal motion (ax = 0).

- These “two” motions (X & Y) are ___________________ & _____________________ (same ____).

- Remember: If YA = YC, then A & C are __________________ (1) ______________ and (2) ______________.

- At any point, the object’s velocity vector is _________________ to its path.

aX = ____ ____ Equation(s):

vAx = _____ = _____ = _____

aY = ____ ____ Equation(s)

For every question: (1) Determine AXIS (X or Y) (2) Pick an INTERVAL (3) Pick an EQUATION

EXAMPLE: An object is launched from the ground with 50 m/s at +37o, and later returns to the ground. Find its:

(a) Maximum height

(b) Time to reach max. height

(c) Total time of flight

(d) Horizontal displacement (range)

● Symmetric launches attain max range when Θ = ____. Complementary angles (eg. ____ & ____) attain the same range.

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PROJECTILE MOTION: HORIZONTAL LAUNCH (Zero Angle)

● In horizontal launches: ___________ _______________.

EXAMPLE 1: An object is launched horizontally with 30 m/s from a 50 m high cliff. Find its:

(a) time to hit the ground.

(b) range (horizontal displacement).

PRACTICE 1: An object rolls from the top of a hill (shown below) with 20 m/s and takes 4 s to hit the floor.

(a) Find the object’s range.

(b) How tall is the hill?

EXAMPLE 2: An airplane is moving horizontally at 500 m above the ground with an unknown speed. A crate released from

the plane travels a horizontal distance of 4,000 m before striking the ground. Find:

(a) the plane’s horizontal speed.

(b) the crate’s final velocity (magnitude & direction).

UAM EQUATIONS

(1) v = vo + a t

(2) v2 = vo2 + 2 a Δx

(3) Δx = vo t + ½ a t2

*(4) Δx = ½ ( vo + v ) t

PROJECTILE MOTION

(1) Determine Axis (X | Y)

(2) Pick an Interval

(3) Pick an Equation

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PRACTICE: HORIZONTAL LAUNCH

PRACTICE 1: When an object that is launched horizontally hits its

target, its velocity has horizontal and vertical components of 80 m/s

and 60 m/s, respectively. Find the object’s range.

PRACTICE 2: In a regulation-sized beer pong table, the ping pong ball is tossed from a horizontal distance of 2.4 meters

and 1.0 meter above the top of its target cup. What horizontal speed must you throw the ball with, so it makes the cup?

PROJECTILE MOTION




UAM EQUATIONS

(1) v = vo + a t

(2) v2 = vo2 + 2 a Δx

(3) Δx = vo t + ½ a t2

*(4) Δx = ½ ( vo + v ) t

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PROJECTILE MOTION: NEGATIVE LAUNCH (Negative Angle)

● In negative launch problems, the object starts off with vertical velocity pointing down.

- These usually have just 1 interval. Note all variables are downward (VY, Δy, aY = g).

EXAMPLE: An object slides off an inclined roof (the angle shown is 37o) with 5 m/s, at 3 m above the ground. Find its:

(a) horizontal distance while in the air.

(b) final velocity (magnitude and direction).

PRACTICE: You throw an object from the top of a building with 50 m/s directed at 53o below the horizontal. It covers a

horizontal distance of 80 m while in the air. Find its:

(a) total time of flight.

(b) final velocity (magnitude and direction).

PROJECTILE MOTION




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PROJECTILE MOTION: LAUNCH UP *FROM* A HEIGHT

● In these problems, the first part of the motion is Symmetric, but the object drops further.

- When picking intervals, try to include point B to simplify equations (_______).

EXAMPLE 1: You throw an object from the top of a 100 m-tall building with 50 m/s directed at 37o above the x-axis. Find:

(a) the maximum height.

(b) the time to reach the maximum height.

(c) the total range (horizontal displacement).

(d) the magnitude & direction of vB, vC, vD.

● If answering multiple questions, it’s often best to break up the motion (___-___, ___-___).

- But sometimes it’s better / easier / faster to solve in a single step / interval (___-___).

EXAMPLE 2: You throw a rock from a 60-m cliff with 50 m/s at 53o above the +x. Using a SINGLE interval, find:

(a) the vertical component of the velocity at D.

(b) the total time of flight.

PROJECTILE MOTION




A

B

C

D

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PROJECTILE MOTION: LAUNCH *TO* A HEIGHT

● In these, we will usually use a combination of the X equation, with the Y equation #3: ΔY = Vot + ½ a t2.

EXAMPLE 1: A fireman, 60 m away from a building, shoots water from a hose at 30 m/s, angled at 53o above the +x. Find:

(a) the time to hit the building.

(b) the height at which it hits the building.

● In some cases, you won’t know if the object hits its target on its way up OR on its way down.

EXAMPLE 2: You want water from hose at 25 m/s and 37o above the +x to reach a building at 10 m above the ground. How

far from the building should the hose be positioned? There are two possible distances, so:

(a) Find the two times it takes for the water to reach the given height.

(b) Find the two distances from the building that would achieve this.

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SYMMETRIC LAUNCH: FIND INITIAL VELOCITY (VA, ΘA, or both) ● Symmetric Launch problems where either VA or ΘA is missing require 2-3 equations. - Remember the vector equations. If you have VAx & VAy, you can find VA & ΘA.

● Y equations in the BC interval (voy = 0) give simple combos of the 3 Y vars: Vy, t, H à - If you have any Y variable, you can use these equations to find the other two.

EXAMPLE: An object is launched at an angle from the ground. It reaches a maximum height of 45 m, and returns to the ground after covering a horizontal distance of 240 m. Find its initial velocity (magnitude and direction).

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PRACTICE: SYMMETRIC LAUNCH: FIND INITIAL VELOCITY

PRACTICE 1: The range of an object launched from the ground at an angle is 300 m. If the vertical component of its

velocity just before returning to the ground has magnitude 50 m/s, find the magnitude and direction of its initial velocity.

PRACTICE 2: An object is launched from the ground with an initial 40 m/s at an unknown angle. It reaches its maximum

height in 3.5 seconds, before returning to the ground. Find its initial launch angle.

PRACTICE 3: An object is launched from the ground with an unknown initial speed directed at 30o above the horizontal. If it

reaches a maximum height of 40 m before returning to the ground, find its initial speed.

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a,C = v,T 2 / r

v,T = __________ = __________

UNIFORM CIRCULAR MOTION

● In Uniform Circular Motion, an object moves with constant speed in a circular path.

v,T = ______________________________

a,C = ______________________________

r = ______________________________

● When an object completes one lap (__________________ or ___________), it covers a distance of _____ = _________.

- Time for one cycle ____________ (___) in [____]

- Inverse of Period ____________ (___) in [____]

Period is seconds/cycle, frequency is cycles/second. RPM: Revs per Minute: f = RPM / 60

EXAMPLE 1: Calculate the period, frequency, and speed of an object moving in uniform circular motion (radius 10 m) if:

(a) it completes 100 cycles in 60 seconds;

(b) it takes 3 minute to complete 1 cycle.

EXAMPLE 2: The car below takes 10 s to go from A to B, at constant speed. If the semi-circle has radius of 5 m, find its:

(a) period; (b) tangential velocity; (c) centripetal acceleration.

NOTE: Even though the object’s speed is constant, its direction changes, therefore its velocity changes and _________.

Constant speed, but NON-ZERO centripetal acceleration (tangential velocity changes direction).

B A

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MORE: UNIFORM CIRCULAR MOTION

PRACTICE 1: A Ping-Pong ball goes in a horizontal circle (radius 5 cm) inside a red cup twice per

second. Find its: (a) period; (b) speed; (c) centripetal acceleration.

EXAMPLE 1: One way to simulate gravity (or “create artificial gravity”) in a space station is to spin it. If a cylindrical space

station (diameter = 500 m) is spun about its central axis, at how many revolutions per minute (rpm) must it turn so that the

outermost points have acceleration equal to the acceleration due to gravity at the surface of the Earth?

PRACTICE 2: A 3kg rock spins horizontally at the end of a 2-m string at 90 rpm. Calculate its: (a) speed; (b) acceleration.

U. CIRCULAR MOTION

a,C = v,T 2 / r

v,T = 2 π r / T = 2 π r f

f = 1 / T = RPM / 60

PHYSICS - CLUTCH


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PHYSICS - CLUTCH CH 03: 2D MOTION (PROJECTILE MOTION)lightcat-files.s3.amazonaws.com/packets/admin_physics-3-clutch-ph… · MOTION IN TWO DIMENSIONS Motion in 2D (plane) in very

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