Motion in One Dimension. Scalar and Vector Quantities Vector- a physical quantity that requires the...

Motion in One Dimension

Scalar and Vector Quantities

• Vector- a physical quantity that requires the specification of both magnitude and direction.

• Scalar- a physical quantity that requires only magnitude.

Displacement• Is the change in position of an object.• Is given by the difference between an

objects final and initial coordinates.• Symbol – Δx

– Where Δ is the symbol for “the change in”

• Δx = xFinal- xinitial

• Units – meter (m)• Vector quantity

Distance• Is how far an object moves.• The path that the object travels matters.• Units – meter (m)• Symbol – d• Scalar quantity

Sign Convention• When dealing with motion in one

dimension, the object only has two direction to travel. These two directions are specified by using + and – signs.

• If the sign of the motion is + the object is moving in the +x direction. Likewise, if the sign of the motion is - the object is moving in the -x direction.

Average Velocity

• Is the displacement, Δx, divided by the time interval during which the displacement occurred.

• Equation:

• Units m/s

Instantaneous Velocity

• The limit of the average velocity as the time interval Δt becomes infinitesimally short.

• Equation:

• Units m/s

Graphical Representation of Average Velocity

• The slope of a position vs. time graph gives the average velocity of an object.

• For any object, the average velocity during the time interval ti to tf is equal to the slope of the straight line joining the initial and final points on a graph of the position of the object plotted vs. time.

Graphical Representation of Instantaneous Velocity

• The instantaneous velocity is defined as the slope of the line tangent to the position-time curve at P.

A toy train moves slowly along a straight portion of track according to the graph of position vs. time to the right.

Find• the average velocity for the total trip.• the average velocity for 0.0 s- 4.0 s. • the average velocity for 4.0 s- 8.0 s.• the average velocity for 8.0 s- 12.0 s.• the instantaneous velocity at t = 2.0 s.• the instantaneous velocity at t = 5.0 s.

Average Acceleration

• The change in velocity during the time interval during which the change occurs.

• Equation:

• Units: m/s2

Instantaneous Acceleration

• The limit of the average acceleration as the time interval Δt becomes infinitesimally short.

• Equation:

• Units m/s2

Acceleration and Velocity

• When the object’s velocity and acceleration are in the same direction, the speed of the object will increase with time.

• When the object’s velocity and acceleration are in opposite directions, the speed of the object will decrease with time.

Motion Maps

One way to describe motion is through a diagram called a motion map. Many different types of motion maps exist, we will start with a simple one.

Graphical Representation of Average Acceleration

• The slope of a velocity vs. time graph gives the average acceleration of an object.

• For any object, the average acceleration during the time interval ti to tf is equal to the slope of the straight line joining the initial and final points on a graph of the velocity of the object plotted vs. time.

Graphical Representation of Instantaneous Acceleration

• The instantaneous acceleration of an object is equal to the slope of the velocity-time graph at that instant in time.

• From now on we will use “acceleration” to mean “average acceleration” .

A baseball player moves in a straight line path in order to catch a fly ball hit into the outfield. His velocity as a function of time is shown in the graph.

A) Find his instantaneous acceleration at points A,B,C on the curve.

B) Describe in everyday language how the outfielder is moving.

Velocity and Acceleration Graphs

• Match the velocity-time graphs with their corresponding acceleration-time graphs.

• Answers :– a e– b d– c f

• Most of what we will be dealing with in this class is constant acceleration.

• Constant acceleration is when the average acceleration is equal to the instantaneous acceleration.

• This means that the objects velocity increases or decreases at the same rate throughout the motion.

ONE DIMENSIONAL MOTION WITH CONSTANT ACCELERATION

Since the average acceleration is equal to instantaneous acceleration, we can rewrite the acceleration as the following:

Unless otherwise specified, let ti =0s. Also for our own convenience, we are going to replace vf with v and vi with vo.

With a little algebra the equation can be rearranged to find the final velocity.

We can write the equation for average velocity as the arithmetic average of the initial velocity (vo) and final velocity (v).

Knowing this previous equation and that

Where xi and ti both equal to 0Then rearrange for x.

So

Plug the equation for average velocity in and you get:

A little algebra and the equation simplifies to

Using Plug in for v

And simplify

Substitute into

And solve for v2

Freefall•Is when an object is moving under the influence of gravity alone. The source of the initial motion is not important.

•Objects that are thrown upward, downward or released from rest are all in freefall once released.

• Once objects are in freefall they have a constant acceleration downward, which is the acceleration due to gravity, g.

• g=9.8m/s2

• g is + or – depending on the definition of the + direction

Freefall Practice Problems

1. A ball is thrown downward from the top of a cliff with an initial speed of 10.0 m/s. Determine the velocity and speed of the ball ay t=2.00s.

Freefall Practice Problems2. A stone is thrown from the top of a building

with an initial velocity of 20.0 m/s upward. The building is 50.0m high, and the stone just misses the edge of the roof on the way down.

Determine• the time to reach the maximum height.• the maximum height.• the time needed to return to the throwers

height.• the velocity of the stone at this height.• the velocity and position of the stone at t =

5.00s.