Kinematics in Dimensions q vy vx x y Projectile Motion Two Kinematics in Dimensions

What is a Projectile? Trajectory A projectile is any object that is placed into free flight and is being affected by gravity. The path of a projectile is called the trajectory. The horizontal distance traveled is called the range. The peak of the projectile path is called the apex. Kick Apex Trajectory Range

More on Trajectories Parabola The trajectory of a projectile forms the shape of a parabola. Parabola

Different Trajectories Depending on initial location and angle, projectiles can form different paths, all of which still possess some parabolic shape. This path is called a “half” parabola. Half Parabola

Different Trajectories Depending on initial location and angle, projectiles can form different paths, all of which still possess some parabolic shape. This path is called a “full” parabola. Full Parabola

Different Trajectories Depending on initial location and angle, projectiles can form different paths, all of which still possess some parabolic shape. This path is called a “partial” parabola. Partial Parabola

Quick Comparison of Paths Half Parabola Launched Horizontally from some height Full Parabola Launched at an angle from ground level Symmetrical trajectory Partial Parabola Launched at an angle from some height Half Parabola Full Parabola Partial Parabola

The Half Parabola v1x Trajectory Mathematically, this is the simplest of all the trajectories. Here an object is launched horizontally and allowed to fall. Because there is no vertical velocity at the onset, v1y = 0. Thus, the time of flight is equal to the time it would take to drop the object from rest. Also note the horizontal velocity vx is constant for all projectiles. Click on the flag for the princess to drop the ball.

Only vertical values will affect time of flight. Half Parabola Timing As stated previously, the time of flight of a half parabolic path is equal to that of simply dropping the object from the same height. Horizontal velocity (vx) has no affect on flight time because it is not affected by gravity. Click on the support post to watch the drop. Only vertical values will affect time of flight.

Sample Problem (Half Parabola) A toy car is raced off a table (1.1m high) onto the floor below. It was traveling at 5.0m/s. How long did it take for the car to crash on the floor? Recall, horizontal and vertical components are independent of one another. Zap!

Sample Problem (Half Parabola)(Cont.) More questions about the problem can be asked. How far from the base of the table did the car travel before crashing? Recall, the time of flight is 0.474s and the initial speed is 5.0m/s. Since horizontal velocity is constant (ax = 0) and no starting position was mentioned (d1x = 0), we can use the equation below. Zap!

Half Parabola Summary Objects must be dropped from some height d1y. The vertical reference point is usually the ground or floor. The time of projectile’s flight is identical for that of simply dropping the object the same distance (straight down). Horizontal velocity remains constant in all projectile problems (v1x = v2x). Final horizontal position calculations reduce to (d2x = vxt).

Initial Velocity Breakdown When an object is launched at some angle, it’s initial velocity (v1) can be broken down into two components. Horizontal Component (Vx) Vertical Component (Vy) What shape is formed? Consider also the launch angle (q). v1 v1y Please Note: horizontal and vertical components are independent of one another. The only commonality is time. Right Triangle q v1x

Initial Velocity Breakdown (Cont.) Consider the breakdown from the previous slide again. There are trigonometric relationships between the sides and angles of a right triangle. Important! v1 v1y q v1x

Practicing Trig Functions Consider the triangle below. Solve for the unknown values. Searching for x Searching for y 13 x y 22.62°

Sample Velocity Breakdown A dart gun is fired at an angle of 30° with a muzzle velocity of 40m/s. Calculate the components of the velocity? Horizontal Component (x) Vertical Component (y) v1 v1y q v1x Dart-X Make sure your calculator is in Degree mode!

The Full Parabola The key to the full parabola is symmetry. Try to identify some points of symmetry. Throw Vy Vx

Full Parabola: Another Look Take another look at the same situation. The peak of trajectory is called the apex. Here vy = 0 because the ball then comes down. The time it takes to reach the apex is half the total. Like any projectile path, vx is constant. We will also find that the initial vertical velocity is the negative opposite of the final (v1y = -v2y) Throw Apex Vy Vx

Sample Full Parabola Problem A golf ball is struck at an angle of q = 37° with the horizontal at a velocity of 45m/s. What are the components of the velocity (v1x and v1y)? Horizontal Component Strike v1 v1y Vertical Component q v1x

Sample Full Parabola Problem A golf ball is struck at an angle of q = 37° with the horizontal at a velocity of 45m/s. How long does it take for the ball to reach the apex? Previously Calculated Fly x y a v1 v2 d1 d2 t Accelerations are always given for projectile problems. Vertically, the ball is at rest at the apex (v2y = 0).

Sample Full Parabola Problem A golf ball is struck at an angle of q = 37° with the horizontal at a velocity of 45m/s. What is the maximum vertical height that the ball reaches? Fly x y a v1 v2 d1 d2 t The ball starts from the ground (d1y = 0) and reaches maximum height at the apex (d2y). The problem could also be solved using the equation

Sample Full Parabola Problem A golf ball is struck at an angle of q = 37° with the horizontal at a velocity of 45m/s. What is the total time of flight? Fly x y a v1 v2 d1 d2 t This is a different situation than when the ball is at the apex. Recall (v1y = -v2y). Notice that the time required is double that to reach the apex.

Sample Full Parabola Problem A golf ball is struck at an angle of q = 37° with the horizontal at a velocity of 45m/s. What is the maximum horizontal distance traveled? Fly x y a v1 v2 d1 d2 t Horizontal velocities are equal (v1x = v2x) because ax = 0. The starting horizontal position is usually zero (d1x = 0).

Sample Problem (Full Parabola) A field hockey ball is struck at an angle of 75° and a velocity of 25m/s. Strike x y a v1 v2 d1 d2 t

Projectile Motion Generator At what angle does the projectile achieve the longest time of flight? At what angle does the projectile travel the farthest distance horizontally? At what angle does the projectile travel the farthest distance vertically? What two angles allow for the object to land at the same place?

Full Parabola Summary Objects end up at the same vertical height they started (d1y = d2y). Horizontal velocity remains constant in all projectile problems (v1x = v2x). The final vertical velocity is the negative opposite of the initial vertical velocity for the entire path. (v2y = -v1y) The time it takes to reach the apex is half the time of flight.

The Partial Parabola Recall, this path has elevation and launch angle. The trajectory again has an apex. This is mathematically the most complex path. Fire

The Partial Parabola If you look at this path carefully, you can see two half parabolas, which simplifies things considerably. You still must consider the launch angle and the components of the velocity when trying to solve. q 1 2

Sample Partial Parabola Problem A cannon nestled in the side of a cliff (d1y = 20m) fires a cannonball at 130m/s at a 40° angle. What are the components of the initial velocity? Horizontal Component Vertical Component Fire

Sample Partial Parabola Problem A cannon nestled in the side of a cliff (d1y = 20m) fires a cannonball at 130m/s at a 40° angle. How long does it take for the cannonball to reach the apex? x y a v1 v2 d1 d2 t List givens. Calculate. Fire

Sample Partial Parabola Problem A cannon nestled in the side of a cliff (d1y = 20m) fires a cannonball at 130m/s at a 40° angle. How high is the apex over the water? x y a v1 v2 d1 d2 t You know d1y. Calculate. Fire