Lesson Title: Forces and Motion

MELC: Describe the horizontal and vertical motions of a projectile.

Content: Projectile Motion

INTRODUCTION:

Uniformly Accelerated Motion

A body is said to have uniform acceleration if it maintains a constant change in its velocity in each time interval along a straight line. This can be along the horizontal (rectilinear) or along the vertical (free fall). For rectilinear motion, let us take a track and field runner competing in the 100-m run as an example. If the runner’s positions are taken at equal time intervals and the change in position for each time interval is increasing, then, the runner is moving faster and faster. This means that the runner is accelerating.

The pull of gravity acts on all objects. So, when you drop something or even when you throw something up, it will go down. Things thrown upward always fall at a constant acceleration which has a magnitude of 9.8 m/s2. This means that the velocity of an object changes by 9.8 m/s every second of fall. Consider a ball thrown upward. As the ball goes up, it decelerates until it stops momentarily and changes direction. That means, it reaches its maximum height before it starts to fall back to the point where it was thrown, and its speed will be equal to the speed at which it was thrown. Note that the magnitudes of the two velocities are equal, but they have opposite directions – velocity is upward when it was thrown, but downward when it returns. Free-fall is an example of uniformly accelerated motion, with its acceleration being -9.8 m/s2, negative because it is downward.

DEVELOPMENT:

Motion in Two Dimensions

Many of the games you play and sporting events you join/officiate in during PE classes involve flying objects or balls. Have you noticed the curved paths they make in mid-air? This curve is what naturally happens when an object, called a projectile, moves in two dimensions –having both horizontal and vertical motion components, acted by gravity only. In physics this is called projectile motion. Not only balls fly when in projectile motion. Have you noticed that in many sports and games, players come “flying” too? Understanding motion in two-dimensions will help you apply the physics of sports and enhance game events experiences.

Projectile motion is a combination of uniform motion along the horizontal and the motion of a freely falling body along the vertical. It is an instance of uniformly accelerated motion in two-dimensions. The moving body is called a projectile, the curved path it travels is known as the trajectory and the horizontal distance it covers is called range. The horizontal and vertical motions of a projectile are completely independent of each other.

Therefore, horizontal and vertical motion can be treated separately.

To answer and submit your learning task 1 digitally, click this link: 👉👉 Learning Task 1 .

ENGAGEMENT:

Note: You will use the equations with sin or cos when the angle is given.

Learning Task 2: PROBLEM-SOLVING:

A marble is thrown horizontally from a tabletop with a velocity of 1.50 m/s. The marble falls 0.70 m away from the table’s edge. How high is the lab table? What is the marble’s velocity just before it hits the floor?

ASSIMILATION:

A school introduced the sports of archery so they can send players to Division and Regional Sports competition in the future. A student tries out. He was taught how to handle the equipment. Then, he sets a bow and arrow and directs it to the target, so the arrow is almost in direct line with the bull’s eye target 30 m away. Will the boy hit a bullseye? Why?

Lesson Title: Forces and Motion

MELC: Investigate the relationship between the angle of release and the height and range of a projectile.

Content: Projectile Motion

INTRODUCTION:

Types of Projectile Motion

I. Projectile Released Horizontally (θ = 0°)

A projectile launched horizontally has no initial vertical velocity. Thus, its vertical motion is identical to that of a dropped object. The downward velocity increases uniformly due to gravity as shown by the vector arrows of increasing lengths. The horizontal velocity is uniform as shown by the identical horizontal vector arrows.

The dashed black line represents the path of the object. The velocity vector v at each point is in the direction of motion and thus is tangent to the path. The velocity vectors are solid arrows, and velocity components are dashed. (A vertically falling object starting at the same point is shown at the left for comparison; vy is the same for the falling object and the projectile.)

For a projectile beginning and ending at the same height, the time it takes a projectile to rise to its highest point equals the time it takes to fall from the highest point back to its original position.

DEVELOPMENT:

Learning Task 1: Figure Analysis

The motion of a projectile may be described in terms of range and the maximum height it reaches. But what affects these kinematic quantities? Below is a figure of the height reached and range covered by a projectile launched at the same initial velocity but at different angles. Study it carefully and answer the questions that follow.

Questions:

1. How do you describe the height reached by the projectile at different projection angles?

2. At what angle does it reach the highest? the lowest?

3. Does the projection angle affect the maximum height reached? Why do you say so?

4. How do you describe the range covered by the projectile at different projection angles?

5. At what angle/s does it reach the farthest? the nearest?

6. What angles cover the same range? What do you call these angles?

7. Does the projection angle affect the range covered? Explain briefly.

ENGAGEMENT:

Learning Task 2: Choose the letter that correctly describes the motion of the basketball at different positions as indicated in the diagram below.

A. vertical velocity is zero; horizontal velocity is constant

B. final velocity is maximum

C. initial velocity is maximum

D. vertical velocity decreases; horizontal velocity is constant

E. vertical velocity increases; horizontal velocity is constant

F. vertical velocity increases; horizontal velocity decreases

G. vertical velocity decreases; horizontal velocity increases

ASSIMILATION:

Supposed you are a javelin throw athlete; how will you apply your knowledge of projectile motion in order to win the event?

Lesson Title: Forces and Motion

MELC: Relate impulse and momentum to collision of objects (e.g., vehicular collision).

Content: Momentum and Impulse

INTRODUCTION:

If the two vehicles in Figure 1 suddenly lose their breaks and crash against the brick wall, which do you think would be more damaging? What factor would affect the impact of collision if their velocities were the same?

In Figure 2, car A (#7) is travelling at 80 km/h while car B (#4) is travelling at 30 km/h. Which of the two cars would be more difficult to stop? Which of the two cars has more momentum?

The baseball player in Figure 3 exerts a great amount of force in hitting the baseball. What is/are the effect/s of this force on the baseball? Why does the man have to do a “follow through” after hitting the baseball?

A body’s momentum is its tendency to resist any change in its state of motion or rest. It is also known as inertia in motion. Momentum depends on two factors, mass, and velocity. For bodies moving at the same velocity, the more massive body has greater inertia in motion therefore has greater momentum. Thus, in Figure 1, if both the truck and the car are moving at the same velocities and lose their brakes, the truck would cause more damage on the wall by virtue of its mass. Two bodies of the same mass but different velocities will also have different momenta. In Figure 2, cars A and B have the same mass. But since car A has greater velocity, it has greater momentum making it more difficult to stop than car B. In physics, an external force acting on an object over a specific time leads to a change in momentum of the object. A special name is given to the product of the force applied and the time interval during which it acts: impulse. As shown in Figure 3, the baseball player exerts force to change the baseball’s momentum by hitting it with a bat. He does a “follow through” to increase his time of contact with the baseball and change its momentum further.

DEVELOPMENT:

The momentum (p) of a body is the product of its mass (m) and its velocity (v), as in the equation: p = mv. Force is needed to change the momentum of a body. This force (F) multiplied by the time of contact (t) is known as impulse (I): I = Ft = Δp. Bodies change their momentum through collisions, which may be elastic or inelastic. The SI unit for momentum (p) and impulse

(I) is newton-second (Ns) or kilogram-meter per second (kg· m/s).

Sample Problems:

1. What is the momentum of a 22-kg grocery cart which travels at 1.2 m/s? Given: m = 22 kg; v = 1.2 m/s

Find: p

Solution:

p = mv

= (22 kg) (1.2 m/s)

= 26.4 kg•m/s

2. An offensive player passes a football of mass 0.42 kg with a velocity of 25.0 m/s due south. If the player is in contact with the ball for 0.050 s, what is the magnitude of the average force he exerts?

Given: m = 0.42 kg; v = 25.0 m/s, south; t = 0.050 s Find: F

Solution:

Ft = mv F = mv

t

= (0.42 kg) (25.0 m/s)

0.050 s

= 210 kg•m/s2

To answer and submit your learning task 1, click this link: 👉👉 Learning Task 1.

ENGAGEMENT:

To answer and submit learning task 2 digitally, click this link: 👉👉 Learning Task 2.

Lesson Title: Forces and Motion

MELC: Infer that the total momentum before and after collision are equal.

Content: Conservation of Momentum

INTRODUCTION:

Momentum Conservation

You have learned before that an external force is required to make an object accelerate. Similarly, if we want to change the momentum of an object, an external force is required. There will be no change in momentum if there is no external force.

Let us take this situation as an example. Two children on skateboards are initially at rest. They push each other so that eventually the boy moves to the right while the girl moves in the opposite direction away from each other. Newton’s Third Law tells us that the force that the girl exerts on the boy and the force that makes the girl move in the other direction are of equal magnitude but opposite direction. The boy and the girl make up a system – a collection of objects that affect one another (Figure 1). No net/unbalanced external force acts on the boy-girl system, thus, the total momentum of the system does not change (Figure 2). Remember that momentum, like velocity and force, is a vector quantity. The momentum gained by the girl is of equal magnitude but opposite direction to the momentum gained by the boy. In this system, no momentum is gained or lost. We say that momentum is conserved.

Figure 1. A system is a group of objects that interact and affect each other. Examples are (a) Bowling ball and pin and (b) two football players.

Figure 2. In this example, the total momentum of the boy- girl system before pushing is zero. After pushing, the total momentum of the boy-girl system is still zero because the momentum of the girl is of equal magnitude but opposite direction to the momentum of the boy. Note that the momentum of the boy alone is not the same before and after pushing; and the momentum of the girl alone is not the same before and after pushing.

DEVELOPMENT:

Objective: Describe how a balloon rocket works and how conservation of momentum explains rocket motion.

Materials: balloon (long shape, if available) drinking straw, string (nylon, if available), and adhesive tape

Procedure:

1. Stretch the string over two posts. You can use chairs or iron stands as posts. Make sure that the string is taut.

2. Inflate the balloon. Twist the open end and temporarily secure it with a paper clip.

3. Tape the straw to the balloon such that it is aligned with the balloon’s opening (see Figure 3).

4. Draw a diagram showing the momentum vectors of your balloon rocket and the air.

Questions:

1. How do these momenta compare?

2. How does the velocity of the air that is pushed out of the rocket compared to the velocity of the balloon rocket?

In this activity, the system at the start, which consists of the balloon and the air inside it are stationary, so the total momentum of the system is zero. When we let the air inside the balloon out, we notice that the balloon moves. The force that causes the balloon to move comes from the air that is pushed out of it. There is no external force involved. Thus, the total momentum of the system is conserved and must remain zero. If the balloon has momentum in one direction, the air must have an equal and opposite momentum for the total momentum to remain zero.

Change in momentum = 0

Total Initial Momentum = Total Final Momentum

0 = pballoon+ pair

- pballoon= pair

- (mv)balloon = - (mv)air

Since the mass of the balloon is greater than the mass of air, the velocity of the air must be greater in magnitude than the velocity of the balloon and must be opposite in direction.

Sample Problem: Two ice skaters stand together. They “push off” and travel directly away from each other, the boy with a velocity of 1.50 m/s. If the boy weighs 735 N and the girl, 490 N, what is the girl’s velocity after they push off? (Consider the ice to be frictionless.)

Given: vboy = 1.50 m/s; wboy = 735 N; wgirl = 490 N

Find: vgirl

Solution:

Remember that w = mg, thus, m = w/g. mboy = wboy/g = 735 N/9.8 m/s2 = 75 kg mgirl = wgirl/g = 490N/9.8 m/s2 = 50 kg

The ice where they stand on is frictionless, thus, no external force is present. The momentum of the boy-girl system is conserved. There is no change in the momentum of the system before and after the push off.

Total Initial Momentum = Total Final Momentum

0 = pboy+ pgirl

- pboy = pgirl

- (mv)boy = (mv)girl

- (75 kg x 1.50 m/s)boy = (50 kg x v)girl

- 37.5 kg m/s = 50 kg (vgirl)

- 37.5 kg m/s = 50 kg (vgirl)

50 kg 50 kg

- 0.75 m/s = vgirl

The girl moves with a velocity of 0.75 m/s opposite to the direction of the boy.

Learning Task 2: Answer the problem below.

A 0.30 kg cart moves on an air track at 1.2 m/s. It collides with and sticks to another cart of mass 500 g, which was stationary before collision. What is the velocity of the combined carts after collision?

ENGAGEMENT:

Elastic and Inelastic Collisions

A collision is an encounter between two objects resulting in exchange of impulse and momentum. Because the time of impact is usually small, the impulse provided by external forces like friction during this time is negligible. If we take the colliding bodies as one system, the momentum of the system is therefore approximately conserved. The total momentum of the system before the collision is equal to the total momentum of the system after the collision.

total momentum before collision = total momentum after collision

Collisions are categorized according to whether the total kinetic energy of the system changes. Kinetic energy may be lost during collisions when (1) it is converted to heat or other forms like binding energy, sound, light (if there is spark), etc. and (2) it is spent in producing deformation or damage, such as when two cars collide. The two types of collision are:

To answer and submit your performance task digitally, click this link: 👉👉 Performance Task.

ASSIMILATION:

Explain how Momentum Conservation is applied in any of the following (choose only one):

A. Ball games (e.g., billiards, bowling, baseball)

B. Contact sports (e.g., boxing, wrestling, mixed martial arts)

C. Theme Park rides/games (e.g., bumper car/boat, air hockey, pin ball)

D. Vehicular accidents