Motion in One Dimension

Motion in One Dimension Golden Valley HS Physics Mr. Campbell

2-1 Displacement and Velocity Motion • Motion is what happens all around us. • Different directions and different speeds. • It requires a special effort to analyze motion as a physicist does.

One-dimensional motion is the simplest form of motion • One way to simplify motion is to consider it only in one direction. • Think of a commuter train on a straight track. • It can only move forward or backward.

Motion takes place over time and depends upon the frame of reference • Describing the motion of a train is simple. • Earth is spinning on an axis, moving around the sun and our sun is moving inside of a galaxy. • To help simplify what we are studying we can use a frame of reference. It is a point that stays fixed. • The gecko’s motion is measured along the x axis using centimeters. Motion

Displacement Displacement is a change in position • The initial position to the final position is called the displacement of the object. • The gecko starts at xi and moves to xf. • The Greek letter (Δ) “delta” shows change. • Displacement can also be shown vertically with the y axis. Displacement

Displacement is not always equal to the distance traveled • If the gecko runs up the tree as shown, the change in displacement is simply yf – yi or 60 cm. • If the gecko moves back down to 50 cm, 50 cm – 20 cm = 30 cm of total displacement (based on the starting point). • If the gecko returns to its starting point its total displacement will be “0”.

Displacement can be positive or negative • In the below example there are only two directions our object can move. • Right will be considered east or positive x. Left will be west or negative x. • Up with be north(positive y) down will be south(negative y)

Velocity • Where an object starts and where it stops does not completely describe its motion. • The ground you are standing on may be moving to the left at 2 cm per year (plate tectonics). • If there is an earthquake, we could move 10 cm in a second!

Average velocity is displacement divided by the time interval • Average velocity is the displacement divided by the time interval. • The S.I. unit is meters per second (m/s) • This value is an average. • You may have stopped for a while, traveled slower or faster for a bit. Velocity

Practice A: Average Velocity and Displacement • During a race on level ground, Andra runs with an average velocity of 6.02 m/s to the east. What is Andra’s displacement after 137 s?  answer 825 m to the east

Velocity is not the same as speed • Speed and velocity sometimes are used interchangeably. • In Physics they are different. • Velocity describes the magnitude of that motion and direction. • Speed however, has no direction only magnitude.

Velocity can be interpreted graphically • Velocity can be plotted on a graph. • Position is normally plotted on the vertical axis and time on the horizontal. • If we have constant velocity, we will have a straight line

This graph shows… • Object 1: positive slope = positive velocity • Object 2: zero slope = zero velocity • Object 3: negative slope = negative velocity Sign Conventions

Instantaneous velocity may not be the same as average velocity • This graph show changing velocity with time. • This causes a curved line on our graph. • Any single point on this graph shows instantaneous velocity. • The instantaneous velocity at a given time can be determined by measuring the slope of the line that is tangent to that point on the position-versus-time graph. • The slope and shape of a graph of acceleration describe the object’s motion.

2.1 Questions1. What is a frame of reference?_____________________2. The Greek letter (Δ) “delta” shows ________3. T / F Displacement is always equal to distance traveled.4. Average velocity is the displacement divided by __________5. Speed has no direction, only __________ The part that does not move change False time magnitude

Sec 2-2 Acceleration Changes in Velocity Acceleration is the rate of change of velocity with respect to time • A bus approaches a stop and begins to apply the brakes. • The speed changes from 9.0 m/s to 0 m/s in over a period of 5 seconds. • If a dog were to run out and the bus stopped in 1.5 s, these two stops would be very different for the passengers.

The rate of change of velocity in a given time is called acceleration. • The magnitude of average acceleration is found by dividing the total change in velocity by the time it takes. • The S.I. unit is the meter per second per second (m/s2) or how much the velocity changes each second. Changes in Velocity

Practice B: Average Acceleration • A shuttle bus slows down with an average acceleration of -1.8 m/s2. How long does it take the bus to slow from 9.0 m/s to a complete stop? Given: vi = 9.00 m/s vf = 0 m/s aavg = -1.8 m/s2  Answer 5.0 s

Acceleration has direction and magnitude • Velocity is positive when a train moves to the right. • If the velocity increase as it travels in this direction, the final velocity will be greater than when it started. • When change in “v” is positive, acceleration is also positive. • On long trips the train may stay at constant velocity for a while. If velocity is not changing Δv = 0 m/s. Therefore acceleration is zero. • When the train slows down, its velocity is still positive but changing. • Because initial velocity is greater than final. Acceleration is negative.

The slope and shape of the graph describe the object’s motion • Looking at this graph, we can see the speed of the train is increasing over time. • “A” shows we have both positive direction and acceleration. • “B” shows positive direction and constant speed – no acceleration. • “C” shows continued positive direction but negative acceleration.

If the train were moving backward (negative direction) and speeding up, we could have negative acceleration also. • When looking at this table, take into consideration which way the train is moving. • If vi is (-) then the train is moving backward.

Motion with Constant Acceleration • This photo was taken with a strobe light flashing every 0.1 second. • Gravity accelerates things as they fall. This object is undergoing constant acceleration. • Since its speed changes with time, it covers a greater distance in each constant time interval.

Displacement depends on acceleration, initial velocity and time • This graph shows an object with constant acceleration is going faster with time. • With constant acceleration, average velocity is equal to the average between initial and final velocity.  • To find displacement, we substitute x and t into the equation. This equation makes finding displacement easier. 

Practice C: Displacement with Constant Acceleration • A racing car reaches a speed of 42 m/s. It then begins a uniform negative acceleration, using its parachute and braking system, and comes to a rest 5.5 seconds later. • Find the distance that the car travels during braking. Answer 120 m

Final velocity depends on initial velocity, acceleration and time • You can use this equation to find final velocity of an object after it has accelerated at a constant rate for any time interval. • For displacement of an object use… • This can be used to find displacement and a certain speed after a displacement has been reached.

Practice D: Velocity and displacement with uniform acceleration • A plane starting from rest at one end of a runway undergoes a uniform acceleration of 4.8 m/s2 for 15 s before takeoff. • What is its speed at takeoff? Answer • How long must the runway be for the plane to be able to take off? Answer 72 m/s 540 m

Final velocity depends on initial velocity, acceleration and displacement • So far our equations for acceleration have required a time interval. • Now we will rearrange the equation, substituting time to find distance or displacement. • This new equation gives us final velocity after a displacement… • When using tis equation, you must take the square root of the right side to find final velocity.

Practice E: Final velocity after any displacement • A person pushing a stroller starts from rest, uniformly accelerating at a rate of 0.500 m/s2. • What is the velocity of the stroller after it has traveled 4.75 m? Given: vi = 0 m/s a = 0.500 m/s2Δx = 4.75 m vf = ?  Answer +2.18 m/s

The equations you have just used are summarized below. • They can also be referenced on P. 58 of you textbook. • If the object starts from rest (vi = 0) you can you the formulas on the right.

2.2 Questions1. Acceleration is the rate of change of _________ with respect to time.2. Acceleration has direction and __________3. When a train slows down, its velocity is still __________ but changing.4. Gravity __________ things as they fall.5. T / F If we have final velocity, initial velocity and acceleration, we can calculate time. velocity magnitude positive accelerates True

Sec 2-3 Falling Objects Free Fall • In 1971, astronaut David Scott showed on the moon (a vacuum), a hammer and a feather will fall at the same rate in the absence of air resistance. • Both objects hit the moon’s surface at the same time.

What goes up must come down Freely falling bodies undergo constant acceleration • In a vacuum chamber, any two objects will fall exactly at the same rate. • In the photo you can see the time interval is the same but the displacement keeps changing. • This shows the objects are accelerating. • This constant acceleration is called free fall. • The magnitude of acceleration due to gravity (g) is -9.81 m/s2. • This acceleration is directed downward, therefore it is negative. • Freely fall constant acceleration

Acceleration is constant during upward and downward motion • The object shown in this strobe light photo goes up, slows, changes direction and falls. • The acceleration on the object is negative the entire time. Even when the object is going up, it experiences a force (gravity) pulling it down. • If it were not for this negative acceleration (gravity), the ball would travel up continuously.

This graph shows gravity’s negative acceleration plotted with change in velocity. • Because acceleration is negative, it continually slopes down to the right. • Objects thrown upward same acceleration as falling.

Freely falling objects always have the same downward acceleration • It may seem confusing to think an object moving up is undergoing a negative acceleration. • When an object is moving down it has negative velocity and negative acceleration, therefore it is going faster with time. • When it is thrown up, it has positive velocity but negative acceleration so it is slowing down.

Practice F: Falling Object • Jason hits a volleyball so that it moves with an initial velocity of 6.0 m/s straight upward. • If the volleyball starts from 2.0 m above the floor, how long will it be in the air before it strikes the floor? Given: vi = +6.0 m/s ag = -9.81 m/s2Δy = -2.0 m  Answer When ball hits the floor, next  -8.7 m/s

…to figure out the overall time the ball was in the air, we use…  Answer This the time it took the ball to go up, go down and then another 2 meters. 1.50 s

2.3 Questions1. In the absence of air, all objects fall at the same ______2. Free falling objects undergo constant ________3. T / F Acceleration is constant during upward and downward motion.4. Objects thrown up have positive velocity but negative acceleration, this causes them to ______ down. rate acceleration True slow

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Motion in One Dimension