Chapter 4 Two-Dimensional Kinematics

Units of Chapter 4 Motion in Two Dimensions Projectile Motion: Basic Equations Zero Launch Angle General Launch Angle Projectile Motion: Key Characteristics

4-1 Motion in Two Dimensions – No Acceleration If velocity is constant, motion is along a straight line:

4-1 Motion in Two Dimensions – Constant Acceleration Motion in the x- and y-directions should be solved separately:

ConcepTest 4.1aFiring Balls I ConcepTest 4.1a Firing Balls I A small cart is rolling at constant velocity on a flat track. It fires a ball straight up into the air as it moves. After it is fired, what happens to the ball? 1) it depends on how fast the cart is moving 2) it falls behind the cart 3) it falls in front of the cart 4) it falls right back into the cart 5) it remains at rest

ConcepTest 4.1aFiring Balls I ConcepTest 4.1a Firing Balls I A small cart is rolling at constant velocity on a flat track. It fires a ball straight up into the air as it moves. After it is fired, what happens to the ball? 1) it depends on how fast the cart is moving 2) it falls behind the cart 3) it falls in front of the cart 4) it falls right back into the cart 5) it remains at rest when viewed from train when viewed from ground vertical same horizontal velocity In the frame of reference of the cart, the ball only has a vertical component of velocity. So it goes up and comes back down. To a ground observer, both the cart and the ball have the same horizontal velocity, so the ball still returns into the cart.

4-2 Projectile Motion: Basic Equations Assumptions: ignore air resistance g = 9.81 m/s 2, downward ignore Earth’s rotation If y-axis points upward, acceleration in x-direction is zero and acceleration in y-direction is m/s 2

4-2 Projectile Motion: Basic Equations The acceleration is independent of the direction of the velocity:

4-2 Projectile Motion: Basic Equations These, then, are the basic equations of projectile motion:

4-3 Zero Launch Angle Launch angle: direction of initial velocity with respect to horizontal

4-3 Zero Launch Angle In this case, the initial velocity in the y-direction is zero. Here are the equations of motion, with x 0 = 0 and y 0 = h:

ConcepTest 4.2 Dropping a Package You drop a package from a plane flying at constant speed in a straight line. Without air resistance, the package will: quickly lag behind the plane while falling 1) quickly lag behind the plane while falling 2) remain vertically under the plane while falling 3) move ahead of the plane while falling 4) not fall at all

You drop a package from a plane flying at constant speed in a straight line. Without air resistance, the package will: quickly lag behind the plane while falling 1) quickly lag behind the plane while falling 2) remain vertically under the plane while falling 3) move ahead of the plane while falling 4) not fall at all same horizontal velocity maintain x-direction Both the plane and the package have the same horizontal velocity at the moment of release. They will maintain this velocity in the x-direction, so they stay aligned. Follow-up: What would happen if air resistance is present? ConcepTest 4.2 Dropping a Package

ConcepTest 4.3aDropping the Ball I From the same height (and at the same time), one ball is dropped and another ball is fired horizontally. Which one will hit the ground first? (1) the “dropped” ball (2) the “fired” ball (3) they both hit at the same time (4) it depends on how hard the ball was fired (5) it depends on the initial height

From the same height (and at the same time), one ball is dropped and another ball is fired horizontally. Which one will hit the ground first? (1) the “dropped” ball (2) the “fired” ball (3) they both hit at the same time (4) it depends on how hard the ball was fired (5) it depends on the initial height They both fall from the same height.Therefore, they will hit the ground at the same time. Both of the balls are falling vertically under the influence of gravity. They both fall from the same height. Therefore, they will hit the ground at the same time. The fact that one is moving horizontally is irrelevant – remember that the x and y motions are completely independent !! Follow-up: Is that also true if there is air resistance? ConcepTest 4.3aDropping the Ball I

ConcepTest 4.3bDropping the Ball II In the previous problem, which ball has the greater velocity at ground level? 1) the “dropped” ball 2) the “fired” ball 3) neither – they both have the same velocity on impact 4) it depends on how hard the ball was thrown

In the previous problem, which ball has the greater velocity at ground level? 1) the “dropped” ball 2) the “fired” ball 3) neither – they both have the same velocity on impact 4) it depends on how hard the ball was thrown same vertical velocity “fired” ball has a larger net velocity Both balls have the same vertical velocity when they hit the ground (since they are both acted on by gravity for the same time). However, the “fired” ball also has a horizontal velocity. When you add the two components vectorially, the “fired” ball has a larger net velocity when it hits the ground. ConcepTest 4.3bDropping the Ball II

4-4 General Launch Angle In general, v 0x = v 0 cos θ and v 0y = v 0 sin θ This gives the equations of motion:

4-4 General Launch Angle Snapshots of a trajectory; red dots are at t = 1 s, t = 2 s, and t = 3 s

4-5 Projectile Motion: Key Characteristics Range: the horizontal distance a projectile travels If the initial and final elevation are the same:

4-5 Projectile Motion: Key Characteristics The range is a maximum when θ = 45°:

Symmetry in projectile motion: 4-5 Projectile Motion: Key Characteristics

Which of the 3 punts has the longest hang time? ConcepTest 4.4aPunts I ConcepTest 4.4a Punts I 4) all have the same hang time 123 h

Which of the 3 punts has the longest hang time? 4) all have the same hang time 123 h vertical motion same height same time The time in the air is determined by the vertical motion ! Since all of the punts reach the same height, they all stay in the air for the same time. Follow-up: Which one had the greater initial velocity? ConcepTest 4.4aPunts I ConcepTest 4.4a Punts I

ConcepTest 4.4bPunts II ConcepTest 4.4b Punts II 12 3) both at the same time A battleship simultaneously fires two shells at two enemy submarines. The shells are launched with the same initial velocity. If the shells follow the trajectories shown, which submarine gets hit first ?

12 3) both at the same time A battleship simultaneously fires two shells at two enemy submarines. The shells are launched with the same initial velocity. If the shells follow the trajectories shown, which submarine gets hit first ? The flight time is fixed by the motion in the y-direction. The higher an object goes, the longer it stays in flight. The shell hitting submarine #2 goes less high, therefore it stays in flight for less time than the other shell. Thus, submarine #2 is hit first. Follow-up: Which one traveled the greater distance? ConcepTest 4.4bPunts II ConcepTest 4.4b Punts II

The spring-loaded gun can launch projectiles at different angles with the same launch speed. At what angle should the projectile be launched in order to travel the greatest distance before landing? 1) 15° 2) 30° 3) 45° 4) 60° 5) 75° ConcepTest 4.6 Spring-Loaded Gun

The spring-loaded gun can launch projectiles at different angles with the same launch speed. At what angle should the projectile be launched in order to travel the greatest distance before landing? 1) 15° 2) 30° 3) 45° 4) 60° 5) 75° A steeper angle lets the projectile stay in the air longer, but it does not travel so far because it has a small x-component of velocity. On the other hand, a shallow angle gives a large x-velocity, but the projectile is not in the air for very long. The compromise comes at 45°, although this result is best seen in a calculation of the “range formula” as shown in the textbook. ConcepTest 4.6 Spring-Loaded Gun

Example A golfer hits a shot with an initial speed of 42 m/s at an angle of 35° above the vertical. Neglect air resistance 1.What maximum height does the ball reach? 2.How long does it take to get there? 3.How long does it take to get back down? 4.How far does it travel horizontally? 5.With what velocity (magnitude and direction) will it hit the ground, if it lands at the same elevation it started from?

Summary of Chapter 2 Distance: total length of travel Displacement: change in position Average speed: distance / time Average velocity: displacement / time Instantaneous velocity: average velocity measured over an infinitesimally small time

Summary of Chapter 2 Instantaneous acceleration: average acceleration measured over an infinitesimally small time Average acceleration: change in velocity divided by change in time Deceleration: velocity and acceleration have opposite signs Constant acceleration: equations of motion relate position, velocity, acceleration, and time Freely falling objects: constant acceleration g = 9.81 m/s 2

Summary of Chapter 3 Scalar: number, with appropriate units Vector: quantity with magnitude and direction Vector components: A x = A cos θ, B y = B sin θ Magnitude: A = (A x 2 + A y 2 ) 1/2 Direction: θ = tan -1 (A y / A x ) Graphical vector addition: Place tail of second at head of first; sum points from tail of first to head of last

Component method: add components of individual vectors, then find magnitude and direction Unit vectors are dimensionless and of unit length Position vector points from origin to location Displacement vector points from original position to final position Velocity vector points in direction of motion Acceleration vector points in direction of change of motion Relative motion: v 13 = v 12 + v 23 Summary of Chapter 3

Summary of Chapter 4 Components of motion in the x- and y- directions can be treated independently In projectile motion, the acceleration is –g If the launch angle is zero, the initial velocity has only an x-component The path followed by a projectile is a parabola The range is the horizontal distance the projectile travels