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### University Physics: Mechanics

Ch4. TWO- AND THREE-DIMENSIONAL MOTION

Lecture 5

Dr.-Ing. Erwin Sitompul

http://zitompul.wordpress.com

Homework 4: The Plane

A plane flies 483 km west from city A to city B in 45 min and then 966 km south from city B to city C in 1.5 h.

From the total trip of the plane, determine:

(a) the magnitude of its displacement;

(b) the direction of its displacement;

(c) the magnitude of its average velocity;

(d) the direction of its average velocity;

(e) its average speed.

→

→

Δr2

Δr1

B

A

C

→

Δrtotal

Solution of Homework 4: The PlaneB

A

483 km, 45 min

966 km, 1.5 h

(a) the magnitude of its displacement

(b) the direction of its displacement

C

- Quadrant I

- Quadrant III

Solution of Homework 4: The Plane

(c) the magnitude of its average velocity

(d) the direction of its average velocity

- Quadrant III

(e) its average speed

Projectile Motion

- Projectile motion: a motion in a vertical plane, where the acceleration is always the free-fall acceleration g, which is downward.
- Many sports involve the projectile motion of a ball.
- Besides sports, many acts also involve the projectile motion.

→

Projectile Motion

- Projectile motion consists of horizontal motion and vertical motion, which are independent to each other.
- The horizontal motion has no acceleration (it has a constant velocity).
- The vertical motion is a free fall motion with constant acceleration due to gravitational force.

Projectile Motion

Two Golf Balls

- The vertical motions are quasi-identical.
- The horizontal motions are different.

Projectile Motion Analyzed

The Horizontal Range

Eliminating t,

vx= v0x

vy= –v0y

- This equation is valid if the landing height is identical with the launch height.

Projectile Motion Analyzed

Further examining the equation,

Using the identity

we obtain

R is maximum when sin2θ0 = 1 or θ0 =45°.

- If the launch height and the landing height are the same, then the maximum horizontal range is achieved if the launch angle is 45°.

Symmetry of Position and Speed

v0= 29.4 m/s

- If the initial elevation and final elevation are the same, the velocity of an object at each elevation will be the same in magnitude , but opposite in direction.
- The object’s height and the speed will be symmetrical around the time when the peak position is reached.

Projectile Motion Analyzed

- The launch height and the landing height differ.
- The launch angle 45° does not yield the maximum horizontal distance.

Projectile Motion Analyzed

The Effects of the Air

- Path I: Projectile movement if the air resistance is taken into account
- Path II: Projectile movement if the air resistance is neglected (as in a vacuum)Our calculation along this chapter is based on this assumption

Example: Baseball Pitcher

A pitcher throws a baseball at speed 40 km/h and at angle θ = 30°.

h

(a) Determine the maximum height h of the baseball above the ground.

Example: Baseball Pitcher

A pitcher throws a baseball at speed 40 km/h and at angle θ = 30°.

d

(b) Determine the duration when the baseball is on the air.

(c) Determine the horizontal distance d it travels.

Simulation: How to Fire the Cannon?

v0

9 m

θ

16 m

A cannon is 1.20 m above the ground. You may adjust theinitial speed and the angle of fire of the cannon.

If the target is horizontally 16 m away from the cannon and at 9 m above the ground, how do you set the cannon so that the projectile can hit the target?

Example: Rescue Plane

Released horizontally

A rescue plane flies at 198 km/h and constant height h = 500 m toward a point directly over a victim, where a rescue capsule is to land.

(a) What should be the angle Φ of the pilot’s line of sight to the victim when the capsule release is made?

Example: Rescue Plane

Released horizontally

A rescue plane flies at 198 km/h and constant height h = 500 m toward a point directly over a victim, where a rescue capsule is to land.

(b) As the capsule reaches the water, what is its velocity v in unit-vector notation and in magnitude-angle notation?

→

Unit-vector notation

Magnitude-angle notation

Example: Clever Stuntman

A stuntman plans a spectacular jump from a higher building to a lower one, as can be observed in the next figure.

Can he make the jump and safely reach the lower building?

He cannot make the jump

Time for the stuntman to fall 4.8 m

Horizontal distance jumped by the stuntman in 0.99 s

Homework 5A: Three Point Throw

A basketball player who is 2.00 m tall is standing on the floor 10.0 m from the basket. If he shoots the ball at a 40.0° angle with the horizontal, at what initial speed must he throw so that it goes through the hoop without striking the backboard? The basket height is 3.05 m.

Homework 5B: Docking the Ship

A dart player throws a dart horizontally at 12.4 m/s. The dart hits the board 0.32 m below the height from which it was thrown. How far away is the player from the board?

As a ship is approaching the dock at 45.0 cm/s, an important piece of landing equipment needs to be thrown to it before it can dock. This equipment is thrown at 15.0 m/s at 60.0° above the horizontal from the top of a tower at the edge of the water, 8.75 m above the ship’s deck.

For this equipment to land at the front of the ship, at what distance D from the dock should the ship be when the equipment is thrown? Air resistance can be neglected.

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