PH 201-4A spring 2007PH 201 4A spring 2007

Kinematics in Two DimensionsKinematics in Two Dimensions

Lectures 5,6

Chapter 3 (Cutnell & Johnon, Physics 7th edition)

Kinematics in Two Dimensions1. Displacement, velocity and acceleration

Th di l t Δ i th t d f th i iti l• The displacement Δr is the vector drawn from the initial position at time t0 to the final position at time t

• Average velocity vav

• Instantaneous velocity

• Average acceleration

• Instantaneous acceleration

V – is tangent to the path of the car

Equations of Kinematics in Two DimensionsProblem 12 The initial velocity of a spacecraft is 2650 m/s directed at an angle 30.0 degrees above the x axes. Two engines then fire for a time of 475s. One gives the

ft l ti i th di ti f 6 30 / 2 Th th dspacecraft an acceleration in the +x direction of ax = 6.30 m/s2. The other produces an acceleration in the +y direction of ay = 2.85 m/s2. What is the speed of the spacecraft when the engines shut off?

Two-dimensional motion is viewed as the combination of the separate x and y motions.x and y motions.

The Motion of Projectiles

The motion with constant vertical acceleration and zero horizontalacceleration and zero horizontal acceleration is called projectile of ballistic motion.

The horizontal component of the velocity remains constant, and the vertical component of the velocity decreases, reaches zero at the instant of maximum height, and this becomes gnegative.

The time it takes for the projectile to rise to its maximum height

As the projectile is moving upward it has some positive vertical velocity vy. When it is coming down it has some negative vertical velocity –vy.

At the top of the trajectory, vy = 0 when t = thp j y, y hFrom #3 => 0 = voy – gth

=> The total time the projectile is in the air:p j

The Range of the Projectile

The maximum height of the projectile

The velocity of the projectile as it strikes the ground

The speed of the projectile as it strikes the ground is equal to the original speed of the projectile.

The direction that the velocity vector makes with the ground is

The angle that v makes as it hits the ground is the negative of the original angle.

Problem 13: The punter on a football team tries to kick a football so that it stays in the air for a long “hang time”. If the ball is kicked with an initial velocity of 25.0 m/s at an angle of 60.0 degrees above the ground, what is the “hang time”?

The vertical motion consists of the ball rising for a time t stopping and returning to theThe vertical motion consists of the ball rising for a time t, stopping and returning to the ground in another time t.

Problem 23: A car drives straight off the edge of a cliff that is 54m high. The police at the scene of the accident note that the point of impact is 130m from the base of a cliff. How fast was the car traveling when it went over the cliff?

During this time the car travels a horizontal distance of 130m

Problem 36: A diver springs upward from a board that is three meters above the water. At the instant she contacts the water her speed is 8.90 m/s and her body makes an angle of 75.0 degrees with respect to the horizontal surface of the water. Determine her initial

l it b th it d d di tivelocity, both magnitude and direction

The motion of a ballistic missile can be regarded as the motion of a projectile because along the greatest part of its trajectory the missile is in free fall. Suppose that a missile is to strike a target 1000 km away. What minimum speed must the missile have at the

fbeginning of its trajectory? What maximum height does it reach when launched with this minimum speed? How long does it take to reach the target? For these calculations assume that g = 9.8 m/s2 everywhere along the trajectory.

Problem: A projectile is fired at an initial velocity of 35.0 m/s at an angle of 30.0 degrees above the horizontal from the roof of a building 30.0 m high, as shown. Find

a) The maximum height of the projectile) fb) The time to rise to the top of the trajectory

c) The total time of the projectile in the aird) The velocity of the projectile at the grounde) The range of the projectile

A woman on a bridge 90.0 m high sees a raft floating at a constant speed on the river below. She drops a stone from rest in an attempt to hit the raft. The stone is released when the raft has 6.00 m more to travel before passing under the bridge. The stone hits

f f f f fthe water 2.00 m in front of the raft. Find the speed of the raft.

• During the time t that it takes the stone to fall, the raft travels a • The stone falls a distance y = 90.0 m indistance of 6.00 m – 2.00 m = 4.00 m• So its speed is v = 4.00m/t

The stone falls a distance y 90.0 m in time t given by y=v0t + 1/2gt2

t=√2y/g = √2.00m/9.80m/s2== 4.29s

=> v = 4.00m/4.29s = 0.932 m/s

Problem: A projectile is fired vertically upward from the edge of a straight-walled cliff. The top of the cliff is 118 m above the water. The initial speed of the projectile is 49 m/s.

a) How high will the projectile go?b) How long will it take to reach this height?c) At what time will the projectile hit the water?