Motion and Forces in Two and Three Dimensions -

Ch.7Scalars and Vectors

Velocity VectorsProjectile Motion

Intro to Vectors

• Scalar• A quantity that has only

magnitude but no direction• Examples?

• Vector• A quantity that has both

magnitude and direction• Examples?

Adding Vectors

• Resultant - a vector that is the sum of two or more vectors

• Vectors are added head to tail.

• Vectors can be moved as long as their orientation is maintained.

Adding Vectors

Commutative Property of Vectors

Operations

• Finding the sides of a right triangle.• Pythagorean theorem• a2 + b2 = c2

• Tan Θ = opp / adj• Sin Θ = opp / hyp• Cos θ = adj / hyp

Adding Vectors

• A humming bird flies horizontally for 9.0m and then flies straight up 3.0m, what is the bird’s displacement?

• 9.5 m, 18º above horizontal

Adding Vectors

• A helicopter flies horizontally for 165 m then moves strait down to land 45.0 m below. What is the helicopter’s total displacement?

• 171 m, 15.3º below horizontal

Resolving Vectors

• Components - the horizontal and vertical (x and y) parts that add up to give a resultant vector.• Vectors can be resolved using

the sine and cosine functions.

Resolving Vectors

• The distance from an observer on the plain to the top of a nearby mountain is 5.3 km, and the angle between this line and the horizontal is 8.4º. What is the height of the mountain?

• 770 m

Resolving Vectors

• A rocket travels 113 m at an angle of 82.4º with respect to the ground and to the south. What is the rocket’s horizontal and vertical displacements?

• x=14.9m south, y=112 m up

Adding non-perpendicular vectors

• The vectors must first be resolved into their x and y components.• Add all the x components

and all the y components.• Find the resultant of the

total x and y.

Adding Non-perpendicular vectors

• A bullet travels 850 m and ricochets off a rock. The bullet travels another 640 m but at 36º from its previous direction. What is the total displacement of the bullet?

• 1320m , 15º

Adding Non-perpendicular vectors

• U.S. Highway 212 extends 55 km at 37º north of east between Newell and Mud Butte, SD. It then continues for 66 km due east from Mud Butte to Faith, SD. If you drive along this part of Highway 212 what will be your total displacement?

• 115 km, 17º north of east

Projectile Motion

• Projectile Motion - motion of objects moving in two dimensions under the influence of gravity

• Projectile motion is free fall with initial horizontal velocity.• Horizontal velocity stays constant

through out.• An object dropped straight down will

hit the floor at the same time as an object launched horizontally.

Projectiles - Horizontal

• Vertical motion• ∆y = .5g(∆t)2

• vy,f = g ∆t

• vy,f2 = 2g ∆y

• Horizontal Motion• ∆x = vx ∆t

• vx = vx,i = constant

Projectiles - Horizontal

• A stunt driver launches his car off a cliff at a speed of 20 m/s. He lands in the lake below 2.0 s later. What is the height of the cliff and the horizontal distance the car travels?

Horizontal Projectile

A squirrel on a limb near the top of a tree loses its grip on a nut, so that the nut slips away horizontally at a speed of 10.0 cm/s. If the nut lands at a horizontal distance of 18.6 cm, how high above the ground is the squirrel?

Horizontal Projectile

The longest shot in a golf tournament was made by Mike Austin in 1974. The ball went a distance of 471 m. Suppose the ball was shot horizontally off a cliff at 80.0 m/s. Calculate the height of the cliff.

Projectiles - Angle

• Vertical Motion• ∆y = vi(sin )∆t

+ .5g(∆t)2

• vy,f = vi(sin ) + g∆t

• vy,f2 = vi

2(sin )2 + 2g∆y

• Horizontal Motion• ∆x = vi(cos )∆t

• vx = vi(cos ) = constant

Projectiles - Angle

• A baseball is hit at an angle of 40 above horizontal at a speed of 35m/s, the ball is then caught by a player at the same height from which it was hit.

• How long was the ball in the air?• How far was it hit?• How high did it fly?

• A zookeeper finds an escaped monkey hanging from a light pole. Aiming her tranquilizer gun at the monkey, she kneels 10.0m from the light pole, which is 5.0m high. The tip of her gun is 1.00m above the ground. The monkey tries to trick the zookeeper by dropping a banana and still holding onto the light pole. At the moment the monkey drops the banana the zookeeper fires. If the tranquilizer dart travels at 50.0m/s, will the dart hit the monkey or the banana?