Mon. Jan. 31st 1

PHSX213 class• I hope you had a good weekend.

– I spent a lot of Friday evening annoyed that my car seems to obey Newton’s 1st Law when I was turning on an icy corner !

• HW1 is due now.• Questions from last time ? (1-d kinematics)• Many of you are now signed up for the

online access – please do this as soon as you can.– HW2 is available (only available with

eGradePlus access)– Submissions

• Kinematics in 2-d (Chapter 4)– Projectile motion– Uniform Circular Motion

Mon. Jan. 31st 2

Position vector for a particle

Mon. Jan. 31st 3

Reading Quiz

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A ball is thrown upward at a 45° angle. In the absence of air resistance, the ball follows a

A. tangential curve.B. parabolic curve.C. sine curve.D. linear curve.

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A ball is thrown upward at a 45° angle. In the absence of air resistance, the ball follows a

A. tangential curve.B. parabolic curve.C. sine curve.D. linear curve.

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Check-Point 0

You are throwing a ball straight up in the air.

At the highest point, the ball’s

A. velocity and acceleration are zero.

B. velocity is nonzero but its acceleration is zero.

C. acceleration is nonzero, but its velocity is zero.

D. velocity and acceleration are both nonzero.

Mon. Jan. 31st 7

Check-Point 0

You are throwing a ball straight up in the air.

At the highest point, the ball’s

A. velocity and acceleration are zero.

B. velocity is nonzero but its acceleration is zero.

C. acceleration is nonzero, but its velocity is zero.

D. velocity and acceleration are both nonzero.

The ball reaches its highest point when its velocity is zero. The acceleration due to gravity is constant and never zero.

Mon. Jan. 31st 8

Check-Point 1A person standing at the edge of a cliffthrows one ball straight up and another ballstraight down at the same initial speed. Neglectingair resistance, the ball to hit theground below the cliff with the greater speedis the one initially thrown :

A. upward. B. downward. C. neither—they both hit at the same speed.

Mon. Jan. 31st 9

Check-Point 1A person standing at the edge of a cliffthrows one ball straight up and another ballstraight down at the same initial speed. Neglectingair resistance, the ball to hit theground below the cliff with the greater speedis the one initially thrown :

A. upward. B. downward. C. neither—they both hit at the same speed.

On its descent, an object with initial velocity, v, has a velocity of –v, when it reaches the height from which it was thrown

Mon. Jan. 31st 10

Kinematics: General Case (3-D)

• Reference frame– Origin, (x, y, z)-axes

• Position Vector, r = x i + y j + z k

• Displacement, ≡ r = r2 – r1

• Instantaneous Velocity, v ≡ d r /dt

• Instantaneous Acceleration, a ≡ d v/dt ≡ d2 r /dt2

^ ^ ^→

→ →→

→

→

→

→ →

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Let’s look in more detail at what this 3-D stuff means

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Projectile Motion• (Neglect air resistance)• Launch a projectile at angle, , to

the horizontal, and with initial speed, v0

• Horizontal component of the velocity, vx , experiences NO acceleration (ie. ax = 0).

• Vertical component of the velocity, vy , has acceleration due to gravity (ie. ay = -g ).

• Projectile motion can be analyzed by considering the horizontal and vertical as independent of each other.

• Range = horizontal distance R.

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Check-Point 2

Consider the situation depicted here. A gunis accurately aimed at a dangerous criminalhanging from the gutter of a building. Thetarget is well within the gun’s range, but theinstant the gun is fired and the bullet moveswith a speed vo, the criminal lets go anddrops to the ground. What happens? Thebullet :• 1. hits the criminal regardless of the

value of vo.• 2. hits the criminal only if vo is large

enough.• 3. misses the criminal.

Mon. Jan. 31st 14

Check-Point 2

Consider the situation depicted here. A gunis accurately aimed at a dangerous criminalhanging from the gutter of a building. Thetarget is well within the gun’s range, but theinstant the gun is fired and the bullet moveswith a speed vo, the criminal lets go anddrops to the ground. What happens? Thebullet :• 1. hits the criminal regardless of the

value of vo.• 2. hits the criminal only if vo is large

enough.• 3. misses the criminal.

Why don’t you convince yourself for next time.

Mon. Jan. 31st 15

A hunter points his rifle directly at a coconut that he wishes to shoot off a tree. It so happens that the coconut falls from the tree at the exact instant the hunter pulls the trigger. Consequently,

A. the bullet passes above the coconut.B. the bullet passes beneath the coconut.C. the bullet hits the coconut. D. A situation similar to this wasn’t

discussed in Chapter 4.

Mon. Jan. 31st 16

A hunter points his rifle directly at a coconut that he wishes to shoot off a tree. It so happens that the coconut falls from the tree at the exact instant the hunter pulls the trigger. Consequently,

A. the bullet passes above the coconut.B. the bullet passes beneath the coconut.C. the bullet hits the coconut. D. A situation similar to this wasn’t

discussed in Chapter 4.

Mon. Jan. 31st 17

Projectile Demo

• Car on track with steel ball.

Mon. Jan. 31st 18

Projectile Problems• Projectile (eg. punted football) launched with

speed, v0, at launch angle, • Time to reach its highest point ?

• Coordinates of highest point ?

• Time to hit ground ?

• Range ?

• Velocity components just before landing ?

• What launch conditions for maximum range ?

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Check-Point 3 A battleship simultaneously fires two shells at

enemy ships. If the shells follow the parabolic trajectories shown, which ship gets hit first?

A. A B. B C. both at the same time D. need more information

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Check-Point 3 A battleship simultaneously fires two shells at

enemy ships. If the shells follow the parabolic trajectories shown, which ship gets hit first?

A. A B. B C. both at the same time D. need more information

The time a projectile spends in the air is twice the time taken to fall from its maximum height.

Mon. Jan. 31st 21

Uniform Circular Motion• An object traveling with

constant speed in a circular path IS accelerating (because the velocity is not constant)

• The centripetal acceleration is given by :

• a = v2/r• Period, T, and revolution

frequency definitions.

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Centripetal acceleration derivation

• Note the MINUS sign.

Mon. Jan. 31st 23

On Wednesday

• Relative Motion (end of chapter 4)

• Start Newton’s Laws (chapter 5)