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3.4 Projectile Motion

- The negatives of the components are the components.
- The method of addition is the same as the method of subtraction.

- You can learn how to add vectors.
- Drag onto a graph and change the angle and length.
- The magnitude, angle, and components can be seen in a variety of formats.

- The motion of falling objects is a type of projectile motion in which there is no horizontal movement.

- Motions along the axes are independent and can be analyzed separately.
- The motion of the vertical and horizontal were seen to be independent.
- The key to analyzing two-dimensional projectile motion is to break it into two motions, one along the horizontal axis and the other along the vertical.
- The horizontal axis is called the x- axis and the vertical axis is called the y- axis.
- The magnitudes are s, x, and y.

- If we kept this format, we would call it displacement with components.

- To describe motion, we have to deal with velocity and acceleration and displacement.
- The components must be found along the x- and y-axes.
- We will assume there are no forces other than gravity.
- The components of acceleration are very easy to understand.
- Because gravity is vertical.
- The equations can be used if both accelerations are constant.

- There are components along the horizontal and vertical axes.
- It makes an angle with the horizontal.

- Break the motion into horizontal and vertical components along the x- and y-axes.
- The axes are used.
- The initial values are marked with a subscript 0.

- The motion can be treated as two separate one-dimensional motions, one horizontal and the other vertical.

- The only variable between the motions is time.

- To find the total displacement and velocity, combine the two motions.

- As the object falls towards the Earth again, the vertical velocity increases in magnitude but points in the opposite direction to the initial vertical velocity.

- As the shell reaches its highest point above the ground, the fuse is timed to ignite.

- The analysis method outlined above can be used because air resistance is very low for the unexploded shell.

- We can define and solve for the desired quantities.

- The height is the altitude or vertical position above the starting point.
- The apex is the highest point in any trajectory.

- The highest point in the trajectory of the shell is found to be at a height of 233 m and 125 m away from the ground.

- The component of the initial velocity is in the y-direction.
- The initial angle is given by and the initial velocity is given by.

- The initial velocity is positive, as is the maximum height, but the acceleration due to gravity is negative.
- The maximum height is dependent on the vertical component of the initial velocity, so that any projectile with a 68.6 m/s initial vertical component of velocity will reach a maximum height of 233 m.

- The numbers are reasonable for large fireworks displays, the shells of which do reach such heights before exploding.
- Air resistance is not completely negligible, and so the initial velocity would have to be larger than given to reach the same height.

- There are more than one way to solve the physics problem of time to the highest point.
- The easiest method is to use.

- The final vertical velocity is zero at the highest point.

- This is a good time for large fireworks.
- You will notice a few seconds before the shell explodes when you see the fireworks launch.

- The horizontal velocity is constant because air resistance is negligible.

- In the absence of air resistance, the horizontal motion is constant.
- The fireworks fragments could fall on spectators if the horizontal displacement is not used.
- Air resistance has a big effect on the shell exploding.

- The expression we found for is valid for any projectile motion where air resistance is not very high.

- The maximum height of a projectile depends on the vertical component of the initial velocity.

- When analyzing projectile motion, it is important to set up a coordinate system.
- An origin for the and positions is a part of the coordinate system.

- Positive and negative directions are defined in the directions.

- The positive vertical direction is usually the direction of the object's motion, while the positive horizontal direction is usually the opposite.
- Since it is directed downwards towards the Earth, the vertical acceleration takes a negative value.
- Sometimes it's useful to define the coordinates differently.
- If you are analyzing the motion of a ball thrown from the top of a cliff, it would make sense to define the positive direction downwards since the motion of the ball is solely in the downward direction.
- Take a positive value if this is the case.

- The world's most active volcano is in Hawaii.
- Red-hot rocks and lava are ejected from active volcanoes.
- The rock strikes the side of the volcano at a lower altitude.

- The rock was ejected from the volcano.

- We can solve for the desired quantities if we resolve this two-dimensional motion into two independent one-dimensional motions.
- The time a projectile is in the air is determined by its vertical motion alone.
- The rock is rising and falling at the same time.
- The final velocity is asked for in this example.
- In the first part of the example, the vertical and horizontal results will be recombined to get the final result.

- While the rock is in the air, it rises and falls to a final position that is 20.0 m lower than its starting altitude.

- We discard the negative value of time because it means an event before the start of motion.

- The time for projectile motion is determined by the vertical motion.
- Any projectile that has an initial vertical velocity of 14.3 m/s and lands 20.0 m below its starting altitude will spend 3.86 s in the air.

- We can find the final horizontal and vertical velocities from the information we have now, combined with the angle it makes with the horizontal.
- It's constant so we can solve it at any horizontal location.
- Since we know the initial angle and initial velocity, we chose the starting point.

- The negative angle shows that the velocity is below the horizontal.
- The final altitude is 20.0 m lower than the initial altitude, which is consistent with the fact that the final vertical velocity is negative and therefore downward.

- projectile motion shows that vertical and horizontal motions are independent of each other.
- The first person to fully comprehend this characteristic was Galileo.
- It was used to predict the range of a projectile.
- Galileo was interested in the range of projectiles for military purposes.
- The range of projectiles can shed light on other interesting phenomena.
- We should consider projectile range further.

- The maximum range is obtained for a fixed initial speed.
- This is not true for other conditions.
- The maximum angle is about.
- There are two angles that give the same range for every initial angle.
- The range depends on the value of gravity.
- Alan Shepherd was able to drive a golf ball a long way on the Moon because of the weaker gravity there.

- The proof of this equation is left as an end-of chapter problem, but it does fit the major features of projectile range as described.

- We assume that the range of a projectile on level ground is very small compared to the size of the Earth.
- If the range is large, the Earth will curve away below the projectile and the direction of gravity will change.
- The range is larger than predicted because the projectile has more time to fall than on level ground.
- If the initial speed is good, the projectile goes into the sky.
- It was not possible before centuries.
- The Earth curves away from the object at the same rate as it falls.
- The object does not hit the surface.
- The rotation of the Earth and other aspects of orbital motion will be covered in greater depth later in the text.

- We see that thinking about the range of a projectile can lead to other topics, such as the Earth's position.

- In addition to velocities, we will look at the addition of velocities, which is an important aspect of two-dimensional kinematics and will yield insights beyond the immediate topic.

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