# Projectile Motion Lecturer: Professor Stephen T. Thornton.

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Projectile Motion Lecturer: Professor Stephen T. Thornton

Reading Quiz Both boys jump off the cliff into the water at the same time. Boy 2 has an additional horizontal velocity. Which boy lands in the water first? A) Boy 1. B) Boy 2. C) They land at the same time. D) Cannot tell without further information.

Answer: C The horizontal and vertical motion is independent of each other. Both start with zero velocity in the y-direction and have the same acceleration. Their vertical motion is identical.

Last Time Vectors

Today Two & three dimensional motion Projectile motion Relative motion

is a position vector from the origin to a given point. Displacement vectors do not depend on origin of coordinate system! See next slide.

is a position vector from the origin to a given point. Displacement vectors do not depend on origin of coordinate system!

(drawn from origin to point)

Velocity and Acceleration Vectors for a Particle Moving Along a Winding Path Trajectory

Constant-Acceleration Equations of Motion x = x 0 + v 0x t + ½ a x t 2 v x = v 0x + a x t v x 2 = v 0x 2 + 2a x  x y = y 0 + v 0y t + ½ a y t 2 v y = v 0y + a y t v y 2 = v 0y 2 + 2a y  x Position as a function of time Velocity as a function of time Velocity as a function of position

Two dimensional motion Horizontal and vertical motions are independent! It is that simple. We will find later that we only need to look at the force components along each direction. It is force that causes acceleration.

Velocity, Addition from Position Vector. Suppose the position of an object is given by (a) Determine its velocity and acceleration as a function of time. (b) Determine at time.

Projectile motion Assumptions:  air resistance is ignored for now.  acceleration of gravity is constant and has value g = 9.80 m/s 2.  Earth’s rotation is ignored.

A projectile is an object moving in two dimensions under the influence of Earth's gravity; its path is a parabola.

Water fountains show projectile motion

Set up solutions Choose coordinate y system. a x = 0 and a y =  g Now what will the earlier kinematic equations look like? x a y =  g

Projectile motion equations Don’t memorize these equations!

Conceptual Quiz: Look at the demo (shoot and drop). Two balls are released at the same time by a spring loaded mechanism. Ball A falls straight down. Ball B is propelled out horizontally. Which ball lands first? A) Ball A B) Ball B C) Impossible to determine. D) Balls land at the same time.

Answer D) Balls land at the same time. Both balls feel the same vertical acceleration, -g. Therefore, they drop at the same rate. Of course their horizontal motion is quite different. Do demo.

It can be understood by analyzing the horizontal and vertical motions separately.

The speed in the x-direction is constant; in the y- direction the object moves with constant acceleration g. This photograph shows two balls that start to fall at the same time. The one on the right has an initial speed in the x-direction. It can be seen that vertical positions of the two balls are identical at identical times, while the horizontal position of the yellow ball increases linearly.

Both boys leave at the same time and land at the same time. What is d? d h

Special Extra 3. Work out solution to previous slide. Problem solution video.

Demo: Projectile motion (monkey and the hunter)

If an object is launched at an initial angle of θ 0 with the horizontal, the analysis is similar to boy jumping off cliff except that the initial velocity has a vertical component. Notice speed changes continuously. Surprise demo.

Consider equations for projectile motion as in lab manual. Range, time, etc. These equations are worked out in most textbooks. Work these out yourself. T is time of flight. h

Conceptual Quiz A projectile is launched from the ground at an angle of 30 °. At what point in its trajectory does this projectile have the least speed? A) just after it is launched B)at the highest point in its flight C) just before it hits the ground D) halfway between the ground and the highest point E) speed is always constant

A projectile is launched from the ground at an angle of 30 º. At what point in its trajectory does this projectile have the least speed? A) just after it is launched B) at the highest point in its flight C) just before it hits the ground D) halfway between the ground and the highest point E) speed is always constant smallest highest point y-component of the velocity is zero The speed is smallest at the highest point of its flight path because the y-component of the velocity is zero. Conceptual Quiz

Which of the three punts has the longest hang time? Conceptual Quiz D) all have the same hang time ABC h

Which of the three punts has the longest hang time? D) all have the same hang time ABC h vertical motion same height same time The time in the air is determined by the vertical motion! Because all of the punts reach the same height, they all stay in the air for the same time. Conceptual Quiz

D) all have the same ABC h Follow-up: which one had the greater initial velocity? Conceptual Quiz

D) all have the same ABC h Follow-up: which one had the greater initial velocity? Conceptual Quiz

A battleship simultaneously fires two shells at enemy ships. If the shells follow the parabolic trajectories shown, which ship gets hit first? A) Ship A. B) Ship B. C) Both at the same time. D) Need more information.

The answer is B. Consider the time for the shell to reach its maximum height (or fall from its maximum height). Since shell A goes higher, it takes a longer time than shell B.

Projectile Motion Applet http://physics.bu.edu/~duffy/semester 1/semester1.html Questions: Three Projectiles Maximum height Time of flight Range Monkey and the Hunter http://galileo.phys.virginia.edu/classes /109N/more_stuff/Applets/Projectile Motion/jarapplet.html

Projectiles with Air Resistance

Relative Velocity in Two Dimensions

Relative speed in one dimension is similar in two dimensions except that we must add and subtract velocities as vectors. Each velocity is labeled first with the object, and second with the reference frame in which it has this velocity.

Here, v WS is the velocity of the water in the shore frame, v BS is the velocity of the boat in the shore frame, and v BW is the velocity of the boat in the water frame. The relationship between the three velocities is:

Discuss relativity car demo. Do not do it before conceptual quiz.

Conceptual Quiz 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? A) it depends on how fast the cart is moving B) it falls behind the cart C) it falls in front of the cart D) it falls right back into the cart E) it remains at rest

Conceptual Quiz Answer 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? A) it depends on how fast the cart is moving B) it falls behind the cart C) it falls in front of the cart D) it falls right back into the cart E) 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.

Airplane Drop.The pilot of an airplane traveling 170 km/h wants to drop supplies to flood victims isolated on a patch of land 150 m below. The supplies should be dropped how many seconds before the plane is directly overhead?

Projectile Angle.At what projection angle will the range of a projectile equal its maximum height?

Copyright © 2009 Pearson Education, Inc. Solving Problems Involving Projectile Motion 1.Read the problem carefully, and choose the object(s) you are going to analyze. 2.Draw a diagram. 3.Choose an origin and a coordinate system. 4.Decide on the time interval; this is the same in both directions, and includes only the time the object is moving with constant acceleration g. 5.Examine the x and y motions separately.

Copyright © 2009 Pearson Education, Inc. Solving Problems Involving Projectile Motion 6. List known and unknown quantities. Remember that v x never changes, and that v y = 0 at the highest point. 7. Strategy. Plan how you will proceed. Use the appropriate equations; you may have to combine some of them. 8. Find the solution.