Embedded Files
Chapter 6: Lesson 1 - Projectile Motion with Math
Chapter 6: Lesson 1 - Projectile Motion with Math
What is a projectile?
What is a projectile?
A projectile, which once projected, continues by its own inertia and is influenced only by the downward force of gravity. Here, you can see a free body diagram for a projectile, where the only force acting on the object is the gravitational force.
What is trajectory?
What is trajectory?
A trajectory is the path that a projectile follows. Here, you can see three different types of trajectories, a ball that is dropped, a ball that is thrown straight up, or a ball that is thrown at an angle.
Range, is the horizontal distance that a projectile travels. When you want to practice hitting golf balls, you go to a driving RANGE. This is a place where you try to hit the ball as far horizontally as you can. At a driving range, you don't want to hit the ball vertically.
There are two components to a projectile's motion, a horizontal component (that goes left to right, like the x-axis) and a vertical component (that goes up and down, like the y-axis.). These components are perpendicular to each other, so they are INDEPENDENT of each other. The motion of the horizontal component does not effect the motion of the vertical component and vice versa. We discuss these two components separately.
You can see in the picture, that the horizontal velocity is in blue and is abbreviated with a v for velocity, with a subscript x that tells us the velocity component is in the horizontal direction, like the x-axis. The vertical velocity is in red and is abbreviated with a v for velocity, with a subscript y that tells us the velocity component is in the vertical direction, like the y-axis.
Look at the picture of the cannon. When you look at the vertical path, which is the direction that gravity acts in, you can see that the distance that the cannon ball falls is increasing each second. The cannon ball is accelerating at 9.8 m/s2. The vertical velocity is increasing and the vertical acceleration is constant, 9.8 m/s2.
When you look at the horizontal path, you see that the spacing between the cannon balls are the same. The cannon ball's horizontal velocity is constant. Whatever the horizontal velocity of the cannon ball is when it leaves the cannon will be the horizontal velocity for the entire trajectory of the cannonball. There is no acceleration in the horizontal direction.
To see the actual trajectory of the cannon ball, you can take the horizontal position of the cannon ball at 1 second and the vertical position of the cannon ball at 1 second and add them together. If you continue to do this for every second that the cannon ball is in motion, then you will see the red cannon ball path that is labeled projectile motion.
Look at the picture above. The horizontal velocity of this cannon ball is 20 m/s and doesn't change for the entire time the cannon ball is in the air. The horizontal velocity is constant and the acceleration is 0 m/s2.
The vertical velocity increases at a rate of 9.8 m/s every second. The vertical acceleration is 9.8 m/s2 and the vertical velocity is increasing.
If you look at this picture, you will see that the horizontal velocity vector is the same for the entire trajectory. Again, the horizontal velocity is constant and the acceleration is 0 m/s2.
The vertical velocity is decreasing as the object goes up, is 0 m/s at the top of the trajectory, and is increasing as the object goes down. The acceleration is 9.8 m/s2 for the entire trip up and down.
Check Your Understanding:
Check Your Understanding:
Click on the down arrow when you have your answer to check to see if you are correct.
A boy standing on a balcony drops one ball and throws another with an initial horizontal velocity of 3 m/s. Which of the following statements about the horizontal and vertical motions of the balls is correct? (Neglect air resistance.)
A. The balls fall with a constant vertical velocity and a constant horizontal acceleration.
B. The balls fall with a constant vertical velocity as well as a constant horizontal velocity.
C. The balls fall with a constant vertical acceleration and a constant horizontal velocity.
D. The balls fall with a constant vertical acceleration and an increasing horizontal velocity.
C. The balls fall with a constant vertical acceleration and a constant horizontal velocity.
The constant vertical acceleration is 9.8 m/s2 and the horizontal velocity is the same, so it is constant.
2. Which of the following conditions is met when a projectile reaches its maximum height?
A. Vertical component of the velocity is zero.
B. Vertical component of the velocity is maximum.
C. Horizontal component of the velocity is maximum.
D. Acceleration in the vertical direction is zero.
A. Vertical component of the velocity is zero.
When a projectile reaches the maximum height of its trajectory, the object stops, so the vertical component of the velocity is 0 m/s.
3. Suppose you toss a ball up and catch it while riding in a bus. Why does the ball fall in your hands rather than falling at the place where you tossed it?
Trajectory depends on the frame of reference. For an observer on the ground, when the bus is moving, your hand is also moving with the same velocity as the bus, i.e. the bus, your hand, and the ball will have the same horizontal velocity. Therefore, the ball will follow a trajectory and fall back in your hands.
Equations
Equations
We are going to expand our available math equations in this lesson. It will be helpful to have your equation sheet ready. A copy of the equation sheet is under resource materials on this website and on google classroom. It can also be found HERE.
Old chapter 3 Math, but NOW with horizontal AND vertical motions..
Old chapter 3 Math, but NOW with horizontal AND vertical motions..
The math in this lesson is the same that we learned in chapter 3, but now we are adding a subscript x to show horizontal motion and a subscript y to show vertical motion. This equation shows that the horizontal distance is equal to the horizontal velocity times time.
dx= vxt
The next two equations are for an object falling with constant acceleration, g. These equations are used when you are talking about vertical motion. Remember, horizontal and vertical motion MUST be calculated separately as they are INDEPENDENT to each other. The subscript y stands for vertical motion. Anytime you have vertical motion, you will have gravity.
dy= vyit+(1/2)gt2
vyf= vyi+gt
Click here for the paper that I use in the video. As you watch this video, pause it and try the math on your own. When you have your answer, start the video and see if you are right. It is important to know where YOU make your mistakes so you can correct them for next time!
Remember there are more practice math problems under Chapter 6 Resources. Practice the math and see if your answer matches my answer.
Practice these Math Problems Before Class!
Practice these Math Problems Before Class!
Click on the picture to take you to the practice problems that I will be going over the next time we have class. Remember, you need to try the problems on your own to see where you make your mistakes if you want to learn how to do the math problems on your own.
Mythbusters: Bullet Fired vs Bullet Dropped
Mythbusters: Bullet Fired vs Bullet Dropped
Does a bullet fired from a horizontal gun reach the ground at the same time as a bullet that is dropped? Physics says that gravity is pulling on both bullets at 9.8 m/s2, but is physics always right? Watch this short clip and find out!
You have finished Lesson 1!
You have finished Lesson 1!
Be sure to head over to google classroom and fill out the exit pass.
Report abuse