Lecture 3:

Freely Falling Objects Free fall from rest: Free fall is the motion of an object subject only to the influence of gravity. The acceleration due to gravity is a constant, g. g = 9.8 m/s 2 For free falling objects, assuming your x axis is pointing up, a = -g = -9.8 m/s 2

1-D motion of a vertical projectile

v t a: v t b: v t c: v t d: Question 1:

1-D motion of a vertical projectile v t a: v t b: v t c: v t d: Question 1:

Basic equations

A ball is dropped from a height of 5.0 m. 1.How long does it take to reach the floor? 2.How fast will it be going when it hits? 3.At what height will it be going half this speed?

A ball is dropped from a height of 5.0 m. 1.How long does it take to reach the floor? {t,y} 2.How fast will it be going when it hits the floor? {v,t} 3.At what height will it be going half this speed? {y,v}

A ball is dropped from a height of 7.0 m. 1.How long does it take to reach the floor? 2.How fast will it be going when it hits? 3.At what height will it be going half this speed?

Question 2Free Fall I a) its acceleration is constant everywhere b) at the top of its trajectory c) halfway to the top of its trajectory d) just after it leaves your hand e) just before it returns to your hand on the way down You throw a ball straight up into the air. After it leaves your hand, at what point in its flight does it have the maximum value of acceleration?

The ball is in free fall once it is released. Therefore, it is entirely under the influence of gravity, and the only acceleration it experiences is g, which is constant at all points. Question 2Free Fall I a) its acceleration is constant everywhere b) at the top of its trajectory c) halfway to the top of its trajectory d) just after it leaves your hand e) just before it returns to your hand on the way down You throw a ball straight up into the air. After it leaves your hand, at what point in its flight does it have the maximum value of acceleration?

Question 3 Free Fall II Alice and Bill are at the top of a building. Alice throws her ball downward. Bill simply drops his ball. Which ball has the greater acceleration just after release? a) Alice’s ball b) it depends on how hard the ball was thrown c) neither—they both have the same acceleration d) Bill’s ball v0v0 BillAlice vAvA vBvB

Both balls are in free fall once they are released, therefore they both feel the acceleration due to gravity (g). This acceleration is independent of the initial velocity of the ball. Alice and Bill are at the top of a building. Alice throws her ball downward. Bill simply drops his ball. Which ball has the greater acceleration just after release? a) Alice’s ball b) it depends on how hard the ball was thrown c) neither—they both have the same acceleration d) Bill’s ball v0v0 BillAlice vAvA vBvB Follow-up: which one has the greater velocity when they hit the ground? Question 3 Free Fall II

You drop a rock off a bridge. When the rock has fallen 4 m, you drop a second rock. As the two rocks continue to fall, what happens to their separation? a) the separation increases as they fall b) the separation stays constant at 4 m c) the separation decreases as they fall d) it is impossible to answer without more information Question 4 Throwing Rocks I

At any given time, the first rock always has a greater velocity than the second rock, therefore it will always be increasing its lead as it falls. Thus, the separation will increase. You drop a rock off a bridge. When the rock has fallen 4 m, you drop a second rock. As the two rocks continue to fall, what happens to their separation? a) the separation increases as they fall b) the separation stays constant at 4 m c) the separation decreases as they fall d) it is impossible to answer without more information Question 4 Throwing Rocks I

A hot-air balloon has just lifted off and is rising at the constant rate of 2.0 m/s. Suddenly one of the passengers realizes she has left her camera on the ground. A friend picks it up and tosses it straight upward with an initial speed of 13 m/s. If the passenger is 2.5 m above her friend when the camera is tossed, how high is she when the camera reaches her?

Solution: we know how to get position as function of time balloon camera Find the time when these are equal

Recall: Scalars Versus Vectors Scalar: number with units Example: Mass, temperature, kinetic energy Vector: quantity with magnitude and direction Example: displacement, velocity, acceleration

Vector addition A B C C = A + B

tail-to-head visualization Parallelogram visualization Adding and Subtracting Vectors B A

D = A - B If then D = A +(- B) C = A + B D = A - B -B is equal and opposite to B

If two vectors are given such that A + B = 0, what can you say about the magnitude and direction of vectors A and B ? a) same magnitude, but can be in any direction b) same magnitude, but must be in the same direction c) different magnitudes, but must be in the same direction d) same magnitude, but must be in opposite directions e) different magnitudes, but must be in opposite directions Question 5Vectors I

If two vectors are given such that A + B = 0, what can you say about the magnitude and direction of vectors A and B ? a) same magnitude, but can be in any direction b) same magnitude, but must be in the same direction c) different magnitudes, but must be in the same direction d) same magnitude, but must be in opposite directions e) different magnitudes, but must be in opposite directions The magnitudes must be the same, but one vector must be pointing in the opposite direction of the other in order for the sum to come out to zero. You can prove this with the tip-to-tail method. Question 5Vectors I

The Components of a Vector Can resolve vector into perpendicular components using a two-dimensional coordinate system: characterize a vector using magnitude |r| and direction θ r or by using perpendicular components r x and r y

Calculating vector components Length, angle, and components can be calculated from each other using trigonometry: A 2 = A x 2 + A y 2 A x = A cos θ A y = A sin θ tanθ = A y / A x AxAx AyAy Magnitude (length) of a vector A is |A|, or simply A relationship of magnitudes of a vector and its component

Adding and Subtracting Vectors 1. Find the components of each vector to be added. 2. Add the x- and y-components separately. 3. Find the resultant vector.

Scalar multiplication of a vector Multiplying unit vectors by scalars: the multiplier changes the length, and the sign indicates the direction.

Unit Vectors Unit vectors are dimensionless vectors of unit length. A A x = A x x ^ A y = A y y ^

Question 6Vector Addition Question 6 Vector Addition You are adding vectors of length 20 and 40 units. Of the following choices, only one is a possible result for the magnitude. Which is it? a) 0 b) 18 c) 37 d) 64 e) 100

Question 6Vector Addition Question 6 Vector Addition a) 0 b) 18 c) 37 d) 64 e) 100 minimum opposite20 unitsmaximum aligned60 units The minimum resultant occurs when the vectors are opposite, giving 20 units. The maximum resultant occurs when the vectors are aligned, giving 60 units. Anything in between is also possible for angles between 0° and 180°. You are adding vectors of length 20 and 40 units. Of the following choices, only one is a possible result for the magnitude. Which is it?

Displacement and change in position Position vector points from the origin to a location. The displacement vector points from the original position to the final position.

Average Velocity t1t1 t2t2 Average velocity vector: So is in the same direction as.

Instantaneous velocity vector v is always tangent to the path. Instantaneous t1t1 t2t2

Average Acceleration Average acceleration vector is in the direction of the change in velocity:

2-Dimensional Motion (sections )

A certain vector has x and y components that are equal in magnitude. Which of the following is a possible angle for this vector in a standard x-y coordinate system? a) 30° b) 180° c) 90° d) 60° e) 45° Question 7: Vector Components II Question 7: Vector Components II

A certain vector has x and y components that are equal in magnitude. Which of the following is a possible angle for this vector in a standard x-y coordinate system? a) 30° b) 180° c) 90° d) 60° e) 45° The angle of the vector is given by tan Θ = y/x. Thus, tan Θ = 1 in this case if x and y are equal, which means that the angle must be 45°. Question 7: Vector Components II Question 7: Vector Components II

a) point 1 b) point 2 c) point 3 d) point 4 e) I cannot tell from that graph. Question 8: Acceleration and Velocity Vectors Below is plotted the trajectory of a particle in two dimensions, along with instantaneous velocity and acceleration vectors at 4 points. For which point is the particle speeding up?

a) point 1 b) point 2 c) point 3 d) point 4 e) I cannot tell from that graph. Question 8: Acceleration and Velocity Vectors Below is plotted the trajectory of a particle in two dimensions, along with instantaneous velocity and acceleration vectors at 4 points. For which point is the particle speeding up? At point 4, the acceleration and velocity point in the same direction, so the particle is speeding up

The Components of Velocity Vector Motion along each direction becomes a 1-D problem vxvx vyvy v

Projectile Motion: objects moving under gravity Assumptions: ignore air resistance g = 9.81 m/s 2, downward ignore Earth’s rotation y-axis points upward, x-axis points horizontally acceleration in x-direction is zero Acceleration in y-direction is m/s 2 x y g vyvy vxvx

These, then, are the basic equations of projectile motion:

Launch angle: direction of initial velocity with respect to horizontal

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:

Zero Launch Angle Eliminating t and solving for y as a function of x: This has the form y = a + bx 2, which is the equation of a parabola. The landing point can be found by setting y = 0 and solving for x:

Trajectory of a zero launch-angle projectile horizontal points equally spaced vertical points not equally spaced parabolic y = a + bx 2

ball 1 ball 2 Where will Ball 1 land on the lower surface? a) Ahead (to the left) of Ball 2 b) Behind (to the right) of Ball 2 c) On top of Ball 2 d) impossible to say from the given information Question 9: Drop and not

ball 1 ball 2 Where will Ball 1 land on the lower surface? a) Ahead (to the left) of Ball 2 b) Behind (to the right) of Ball 2 c) On top of Ball 2 d) impossible to say from the given information Question 9: Drop and not

x y vxvx vxvx v y =0

x y ΔxΔx vxvx vxvx vyvy v

x y v

A ball is projected horizontally at the same time as one is dropped from the same height. Which will hit the floor first?

-g v 0 Sin(θ) v 0 Cos(θ) General Launch Angle In general, v 0x = v 0 cos θ and v 0y = v 0 sin θ This gives the equations of motion:

Range: the horizontal distance a projectile travels As before, use and Eliminate t and solve for x when y=0 (y = 0 at landing)

Relative Motion The speed of the passenger with respect to the ground depends on the relative directions of the passenger’s and train’s speeds: Velocity vectors can add, just like displacement vectors

Relative Motion This also works in two dimensions:

You are riding on a Jet Ski at an angle of 35° upstream on a river flowing with a speed of 2.8 m/s. If your velocity relative to the ground is 9.5 m/s at an angle of 20.0° upstream, what is the speed of the Jet Ski relative to the water? (Note: Angles are measured relative to the x axis shown.)

Now suppose the Jet Ski is moving at a speed of 12 m/s relative to the water. (a) At what angle must you point the Jet Ski if your velocity relative to the ground is to be perpendicular to the shore of the river? (b) If you increase the speed of the Jet Ski relative to the water, does the angle in part (a) increase, decrease, or stay the same? Explain. (Note: Angles are measured relative to the x axis shown.)

Question 11: Relativity Car 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

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 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. Question 11: Relativity Car

 Assignment 1 on MasteringPhysics due Friday, September 6 (12:59 pm).  Exit using the rear doors!