3.3 Projectile Motion

First off, thank you so much for those who have continued to follow my blog. I have been a busy with trying to get school going among other things.


Finally we get to use all these vectors we have been going over the past few weeks. First here are the answers from yesterday:

The question:
A jet plane is flying at a constant altitude. At time t1 = 0 it has components of velocity Vx = 90m/s,  Vy = 40m/s. a) Sketch the velocity vectors at t1 and t2. At time t2 = 30.0 s the components are vx = -170 m/s, vy = 40 m/s
 How do these two vectors differ? For this time interval calculate (b) the components of the average acceleration, and (c) the magnitude and direction of the average acceleration.

Projectile Motion

A projectile is any object that follows a path that is completely determined by gravitational acceleration and air resistance, all the while giving an initial velocity to start. A projectile that is following a path is also known as a trajectory. 

Try this example for fun: Grab to items of the same weight (about) and hold them in your hand at the same height. Next drop one of the objects and throw the other object a few inches out. Which one will reach the ground first? Trick question ;)

They should reach the ground around the same time. Something that is very interesting when I first started learning physics. As of right now we will not include air resistance in our equations for projectile motion.


Heres another question for you to figure out: (Will post tonight or tomorrow morning)

A batter hits a baseball so that it leaves the bat at speed v0 = 37.0 m>s at an angle a0 = 53.1°. (a) Find the position of the ball and its velocity (magnitude and direction) at t = 2.00 s. (b) Find the time when the ball reaches the highest point of its flight, and its height h at this time. (c) Find the horizontal range R—that is, the horizontal distance from the starting point to where the ball hits the ground.

 

3.2 Acceleration Vector

Acceleration describes how a velocity of a particle changes over time. So our equation is very similar to the velocity vector equation. It is (V2-V1)/(time2-time1) basically the change in velocity over the change in time.

Now for those of you who know calculus, instantaneous acceleration at any point is the limit of the average acceleration vector when x2 approaches x.

The acceleration is tangent to the path on if the particle moves in a straight line.

This is an example from University Physics text book, "to convince you that a particle has a nonzero acceleration when moving on a curved path with a constant speed, think of your sensations when you ride in a car. When the car accelerates, you tend to move inside the car in a direction opposite to the car's acceleration. Thus you tend to slide toward the back of the car when it accelerates forward and toward the front of the car when it accelerates backward. If the car makes a turn on a level road, you tend to slide toward the outside of the turn; hence the car has an acceleration toward the inside of the turn."

There is always an equal and opposite reaction.

Having said that, lets do a practice problem from University Physics with acceleration:

A jet plane is flying at a constant altitude. At time t1 = 0 it has components of velocity Vx = 90m/s, Vy = 110m/s. At time t2 = 30seconds the components are Vx = -170m/s, Vy = 40m/s. a) Sketch the velocity vectors at t1 and t2. How do these two vectors differ? For this time interval calculate (b) the components of the average acceleration, and (c) the magnitude and direction of the average acceleration.

Will post answers 1/17

Mark Jackson
Physics/Political Science Student
Metropolitan State University of Denver

3.1 Position and Velocity Vectors

Answers to the review questions for the end of Chapter 2. To enlarge the image, all you need to do is click on it.

So I have been out of the game for a little while due to some school work that needed to be done. But now I am back and ready to work.

We are now onto another basic yet fundamental function of physics. Position, velocity, and acceleration vectors.

Position vectors relate to a particle at a point in time (pretty basic huh?)

Velocity vectors have to do with a particle moving toward one position to another over a change in time. Basically it is the change in position over a change in time.

However velocity has a little twist to it. This is twist is called instantaneous velocity. Instantaneous velocity represents the limit of the average velocity as it approaches zero in relation to the change in position in time.

I will post the next set of lecture slides late today on acceleration vectors.



Mark Jackson
Physics/Political Science student
Metropolitan State University of Denver

Answers to 2.5; review questions for ch. 2

So here are my answers from the previous post. I made these answers at midnight last night. If your cannot read any of the numbers or words please comment.

And now here are the next questions. Courtesy of University Physics

1) You normally drive on the freeway between San Diego and Los Angeles at an average speed of 105 km>h 65 mi>h, and the trip takes 2 h and 20 min. On a Friday afternoon, however, heavy traffic slows you down and you drive the same distance at an average speed of only 70 km>h 43 mi>h. How much longer does the trip take? 

2) The fastest measured pitched baseball left the pitcher’s hand at a speed of 45.0 m>s. If the pitcher was in contact with the ball over a distance of 1.50 m and produced constant acceleration, (a) what acceleration did he give the ball, and (b) how much time did it take him to pitch it?

3) You throw a glob of putty straight up toward the ceiling, which is 3.60 m above the point where the putty leaves your hand. The initial speed of the putty as it leaves your hand is 9.50 m>s. (a) What is the speed of the putty just before it strikes the ceiling? (b) How much time from when it leaves your hand does it take the putty to reach the ceiling?

  

2.5 Free Falling Bodies

There is an acceleration that we deal with every single day. Gravity. We will use this for our everyday practices in the mechanics portion of physics. So this section we will discuss various problems and then revisit them either tomorrow (7/26) or (7/27). This will be the last section before the end of chapter 2. We will begin chapter 3, for sure, on Sunday July 28th.

1) A one-euro coin is dropped from the Leaning Tower of Pisa and falls freely from rest. What are its position and velocity after 1.0s, 2.0s, and 4.0s?

2) You throw a ball vertically upward from the roof of a tall building. The ball leaves your hand at a point even with the roof railing with an upward speed of 15.0 m/s; the ball is then in free fall. On its way back down, it just misses the railing. Find (a) the ball’s position and velocity 1.00 s and 4.00 s after leaving your hand; (b) the ball’s velocity when it is 5.00 m above the railing; (c) the maximum height reached; (d) the ball’s acceleration when it is at its maximum height.

Use the equations we learned in the last section. Again we will revisit these problems on 7/27

Mark Jackson
Political Science/Physics Student
Metropolitan State University of Denver

2.4 Constant Acceleration

First off constant acceleration means, as it states, your acceleration will stay the same no matter what. However,  does this mean that our velocity will stay the same too? No. In fact, our velocity will be increasing, either in the positive direction or in the negative direction.

So the equation for constant x-acceleration we should know is:
x-velocity = (x-velocity at time = 0) + (the product of x-acceleration and the time interval)

I do not have any equation tools on Blogger so we will have to stick to this.

Another equation is when the constant acceleration has a x-velocity that changes at a constant rate.
This equation will be expressed as

average x-velocity = (initial velocity + later time velocity)/ (2)

However this equation will fail if the velocity changes over time. The equation we must express is:
average x-velocity = initial velocity + .5(the product of x-acceleration and the time interval)

Seems pretty straight forward but anyone can screw this up quick. Pay attention to details. Now we can combine all of these equations to form one equation..

x = initial position + (the product of initial velocity and a later  time) + ((.5) (the product of acceleration and time ^2)

Here are some other equations that relate to constant acceleration. Each equation has something different about it. One may include time and not include acceleration. You must distinguish between the two when doing problems.

V_x² = V_0x² +2a_x(x-x_o)

x-x_o = ((V_ox + V_x ) / (2))

Average and Instantaneous Acceleration 2.3 -- Answer to Question 2.2

So first off, here is our answer to the question from the previous section -------------------------------------->>

We will likely have another question as soon as we finish 2.4 which will likely be tomorrow the 24th.

Now onto Average and Instantaneous Acceleration. Similar to instantaneous velocity, but not quite.

Instead of using the distance between the two objects, average velocity, we use the velocities between two time intervals. It is basically the change in velocities over the change in time.

Now instantaneous velocity is also similar to instantaneous acceleration. Instantaneous acceleration is the derivative of velocity with the respect to time.

I was going to post a fantastic graph but I cannot get it to copy and past correctly onto the is blog. So I will explain it.

The greater the curvature (upward or downward) of an object’s x-t graph, the greater is the object’s acceleration in the positive or negative x-direction.

Hopefully that makes some sense. Honestly the graphing part is not as important as the mathematical part of physics. Once you understand how the graphs work, you will understand the majority of graphs you will look at through physics.

Next topic will be on constant acceleration and free falling bodies. BOTH important topics

Mark Jackson