Equations of Motion

Equations of Motion

v2 v1 at

d v1t 12

at 2

d v2t 1

2at 2

d v1 v2

2

t

v22 v1

2 2ad

Uniform Motion

d = vt

Uniform Acceleration

Acceleration due to Gravity

• Close to surface of a large body the acceleration due to gravity is constant– gEarth =

– gMoon =

• Air resistance depends on

• Terminal Velocity is reached when

Ex. 1

• What factors affect an object’s terminal speed? How does each factor affect the terminal speed?

• Does the concept of terminal speed apply on the Moon? Why or why not?

Ex. 2• Sketch a graph of the vertical speed as a function of

time for a skydiver who jumps from an aircraft, reaches terminal speed, opens the parachute, and reaches a new terminal speed.

Ex. 3

• During the first minute of blastoff, a space shuttle has an average acceleration of 5g (i.e., five times the magnitude of the acceleration due to gravity on the surface of Earth).

Calculate the shuttle’s speed in metres per second and kilometres per hour after 1.0 min. (These values are approximate.)

Ex. 4• A flowerpot is dropped from the balcony of an

apartment, 28.5 m above the ground. At a time of 1.00 s after the pot is dropped, a ball is thrown vertically downward from the balcony one storey below, 26.0 m above the ground. The initial velocity of the ball is 12.0 m/s [down].

Does the ball pass the flowerpot before striking the ground?

If so, how far above the ground are the two objects when the ball passes the flowerpot?

Graphical Analysis

Work on Graph Worksheet

1. Using the graph, determine each of the following.

• (a) the average velocity for the first 2.5 s• (b) the instantaneous velocity at the points B. • (c) the times at which the instantaneous velocity is

approximately zero.

2. A ball rolls along the floor, up a sloping board, and then back down the board and across the floor again. The graph below

represents this motion.

• (a) At what time is the ball at its highest point?

• (b) What was the acceleration when the ball was (i) rolling up the board, (ii) rolling down the board, and (iii) at rest at the top point?

• (c) How far up the board did the ball go?

• (d) What was the total displacement of the ball over the 9.0 s trip?

(e) Draw the corresponding position-time graph.

Vectors in 2-D

Vector Components : In General

θ

Define θ at its base to the horizontal

For any vector with magnitude |v|

|v|

Vector Components (cont.)

• Any vector can be written as the sum of perpendicular unit vectors– Standard Form: Magnitude [ N of W]– Component Form:

(Magnitude x + Magnitude y) Units

30 km [30 E of N]d ��������������

Ex.

Adding Vectors

• We add vectors using the tip-to-tail method

Adding Vectors (Cont.)Given two vectors that we are adding

What is their resultanti.e. v1 + v2 = v3

What is the magnitudeand direction of v3 ?

Add up all the horizontal components

Add up all the vertical components

Then use Pyth. Theorem and trig to find Magnitude of v3 and direction (ө)

Then use Pyth. Theorem and trig to find Magnitude of v3 and direction (ө)

ө

Ex. 1

• While out shopping in New York, Melissa got lost. Tracing back her steps, she had walked 100 m [SE], and 600 m [W20°S]. The whole trip took 1.3 hours.

– Determine the distance traveled and resultant displacement.

– Determine her average speed and velocity.

Ex. 2

• A ball rolling with an initial velocity of 30 m/s [W] undergoes an acceleration of 8.0 m/s2[N] for a period of 8.0 seconds.(a) What is the final velocity of the ball?(b) What is the displacement of the ball in the 8.0 s?

Ex. 3

• Jim is 200 m[S] of Mary. Mary begins to walk East at 3.00 m/s the same time that Jim begins to walk West at 4.00 m/s. What is the displacement of Jim from Mary 50.0 seconds later?

SPH4U – RELATIVE VELOCITY

For each of the following, perform the vector operation indicated to find either the sum or the difference vector.

Relative VelocitiesFrames of Reference• A frame of reference is a coordinate systems in which

motion can be described• All velocities are measured relative to a frame of reference• Different frames of reference will describe motion in

different ways• We can easily move between frames that are moving at

constant velocities

v AC

v AB

v BC

v AB v BA

Moving between frames

PG PA AGv v v In Air:

vPG = velocity of plane w/ wind

vPA = velocity of plane w/ no wind

vAG = velocity of wind

SG SW WGv v v

In Water:

vSG = velocity of swimmer w/ current

vSW = velocity of swimmer w/ no currentvWG = velocity of current

Ex. 2: (pg 56 # 4)

A plane, travelling with a velocity relative to the air of 320 km/h [28° S of W], passes over Winnipeg. The wind velocity is 72 km/h from the North. Determine the displacement of the plane from Winnipeg 2.0 h later.

7.2 x 102 km [38º S of W] from Winnipegd

Projectile Motion

Projectile Motion

• A projectile is an object that only moves under the influence of gravity

• A trajectory is the path a projectile takes through the air

• The vertical and horizontal components of motion are independent

Symmetric Trajectories

• Ex. What angle should you fire for maximum time of flight?

• Ex. What angle should you fire for maximum range?

• Ex. What angle should you fire for maximum height?

• Ex. Can two firing angles result in the same horizontal range? Why?

Analysis of Projectile MotionHorizontal (x) Vertical (y)

- Uniform Motion (ignoring air resistance)

- v1x = v1cosӨ

- v2x = v1x

- ∆t is the same in both components- dx = v1x ∆t

- Uniform acceleration- Acceleration due to gravity, ay = g

- v1y = v1sinӨ

- ∆dy = change in height (up = +ve, down = -ve)

- Use kinematics equations to find ∆t and v2y

2 22 2 2x yv v v

1tan y

x

vBelowHorizontal

v

Ex 1:

• A marble rolls off a table with a velocity of 1.93 m/s [horizontally]. The tabletop is 76.5 cm above the floor. If air resistance is negligible, determine– (a) how long the marble is airborne– (b) the horizontal range– (c) the velocity at impact

Ex 2:• A seagull with a rocket strapped on its back is

diving towards its target at an angle of 45o below the horizontal and at a speed of 320 m/s. When the seagull is 600 m above the ground, it releases its load, which then hits the target.

– How long before the “bomb” hits its target?– What horizontal distance will the “bomb” travel?– What final velocity does the “bomb” hit the target?

Projectile Motion(2 objects)

• During the 2013 World Series between the Cardinals and the Red Sox, a ball was batted from home plate, 1.00 m above ground, at v1

km/h, θ degrees above the horizontal. An outfielder, standing 70.0m from home plate, accelerated from rest at 3.60m/s2 for 3.00s to catch the ball when it returned to its initial height. Determine the initial velocity of the ball.