Movement in one dimension.

August 2012

Objectives. To describe motion in terms of

displacement, time, acceleration and velocity.

To interpret graphs of position vs. time.

To apply kinematic equations to calculate distance, time or velocity under conditions of constant acceleration.

To relate the motion of free falling objects to motion with constant acceleration.

DisplacementA displacement, ∆x of an object is defined as its change in position.It is given by de formula:

∆x = -

VelocitySpeed and velocity are used in day to day conversations as interchangeable. But in physics, Speed is a scalar quantity having only magnitude, and Velocity is a vector having magnitude and direction.

The average speed of an object over a given time interval is defined as:

Average speed =

The average velocity of an object is a vector quantity and is defined as:

=

Instantaneous velocity is the limit of the average velocity as the time interval becomes very small.

v ≡

Did YOU know?

The branch of physics concerned with motion and forces is called mechanics. The subset of mechanics that describes motion without regard to its causes is called kinematics.

Activity: Plane schedule.In teams search for different plane schedules and calculate the average velocities of a plane flying the same route in opposite directions.

Hypothesize about why there is a difference.

Share your findings with the rest of the group.

Activity: Case.Your team of experts has been assigned with investigating the recent accident of a train in Spain.

You have the following information:

A video of the accident http://www.youtube.com/watch?v=5HgijA7zz-c

The train was a RENFE Class 130

Your mission to prove if the rumor that the train was going at twice the allowed speed (80km/h) on that part of the track are true.

Post your findings on the Socrative exit ticket.

Problem Solving Strategy• Read and reread the problem carefully before trying to solve it.• Decide what objects or objects you are going to study and for what time

interval. You can often choose the initial time to be t=0.• Draw a diagram or picture of the situation. Usually we choose the x axis

to the right as positive.• Write down the known or given quantities the what you want to know.• Think about which principles of physics apply in the problem.• Consider which equations relate to the quantities involved.• Carry out the calculation.• Think carefully about the result obtained: Is it reasonable? Does it make

sense according to your own intuition and experience?• Keep track of the units. If the units do not balance, a mistake has no

doubt been made.

Examples• The position of a runner as a function of time is plotted as moving along the

x axis of a coordinate system. During a 3.00 second time interval, the runner’s position changes from x1= 50.00m. To x2= 30.5m. What is the runner’s average velocity?

Solution: -6.50 m/s

• How far can a cyclist travel in 2.5h along a straight road if her average velocity is 18 km/h?

Solution: 45 km.

• If you are driving 110km/h along a straight road and you look to the side for 2.0 seconds, how far do you travel during the inattentive period?

Solution: 61.11 m.

• A bowling ball traveling with constant speed hits the pins at the end of the bowling lane 16.5m long. The bowler hears the sound of the ball hitting the pins 2.50 seconds after the ball is released from his hands. What is the speed of the ball? Consider speed of sound as 340 m/s

Solution: 6.73 m/s

AccelerationAn object whose velocity is changing is said to be accelerating.Acceleration specifies how rapidly the velocity of an object is changing.

Average acceleration the is expressed as:

Average acceleration =

=

Motion at constant Acceleration

Let’s define our initial time as zero.

Then: =

A common problem is to determine the velocity of an object after any elapsed time when we are given the object constant acceleration. Solving for in the last equation then:

V

ExampleThe acceleration of a motorcycle is known as 4.0 m/and we wish to determine how fast it will be going after 6 seconds.

V = + at = 0 + (4.0 m/)(6 s) = 24 m/s

Now let’s see how to calculate the position of an object after a time when the object has constant acceleration.

From average velocity we have that: x =

Because the velocity increases at a uniform rate the average velocity will be the midway between the initial and final velocities.

V =

Combining the last to equations we get :

x =

We can also derive another equation which is useful in situations where the time is unknown.

ExampleYou are designing an airport for small planes. One kind of airplane must reach a speed before take off of at least 27.8 m/s (100km/h) and can accelerate at 2.00 m/ (a) If the runway is 150 m long, can the airplane reach the required speed for take off? (b) If not, what minimum length must the runway have?

We know: , , x = 150m, a = 2.00 m/We want: v

=)= 24.5m/s

This runway is not sufficient.Using the same equation we need to find = = 193m.A 200m runway is more appropriate for the plane.

Practice• How long does it take a car to cross a 30.0 m wide intersection after the

light turns green, if the car accelerates from rest at a constant 2.00m/.

Solution: 5.48 seconds

• A car slows down uniformly from a speed of 21 m/s to rest in 6 seconds. How far did it travel in that time?

Solution: 63m.

• A drag racer starts her car from rest and accelerates at 10.0 m/ for a distance of 400m. A) How long did it take the race car to travel this distance? B) What is the speed of the race car at the end of the run?

Solution: 8.9 seconds 89.44 m/s

• A car starts from rest and travels 5 seconds with uniform acceleration of +1.5m/ The driver then applies the brakes, causing a uniform acceleration of -2.0m/ If the brakes are applied for 3 seconds A) How fast is the car going at the end of the breaking period? B) How far has the car gone?

Solution: 1.5m/s 50m.

Free FallingWhen air resistance is negligible, all objects dropped under the influence of gravity near Earth’s surface fall towards Earth with a constant acceleration.

Gravity acceleration is expressed as:

g = 9.8 m/

Galileo formulated the laws that govern the motion of objects in free fall.

Experiment

A fast flash photograph shows two objects of different weight dropped at the same time and falling with the same acceleration.

Practice

If Batman drops from a 10.5 m height. Find the following information:

a) Batman’s total flight time.

b) Batman’s velocity when he hits the speeding car.

c) Batman’s velocity at mid flight.

• Supose a ball is thrown from a tower 70m high with an initial velocity of 3m/s. a) What will be the position after 1s, 2s? b) What would be the speed after 1s and 2s?

Solution: 7.90m, 25.6m 12.8 m/s, 22.6m/s

• A person throws a ball upward into the air with an initial velocity of 15m/s. Calculate a) how high it goes, b) how long the ball is in the air before it comes back to his hand, c) how much time it takes for the ball to reach the maximum height, and d) the velocity of the ball when it returns to the thrower’s hand.

Solution: 11.5m 3.06s 1.53s -15m/s

Interesting Fact

The acceleration of objects such as rockets and fast airplanes is often given as a multiple of g. For example a plane pulling out of a dive an undergoing 3.00g’s would have an acceleration of (3.00)(9.8 m/) = 29.4 m/

Graphical Analysis of Linear Motion

The time t is considered the independent variable and is measured along the horizontal axis. The position x, the dependent variable and is measured along the vertical axis.

The slope of the x vs. t graph is equal to the velocity.

Similarly, the slope at any point of an v vs. t graph equals the acceleration at that moment.

References• Giancoli , Douglas C. Physics Sixth Edition. USA Pearson 2005

• Serway, Raymond A. Essentials of College Physics. USA Thomson 2007

• http://www.youtube.com/watch?v=_Kv-U5tjNCY

• http://www.youtube.com/watch?v=3BETTsLOgQI&feature=related