Activity 2.2

Group Number:       _________________

Group Members:     _________________, ___________________, __________________, ____________________ ,

                                    _________________, ___________________, __________________, ____________________ ,

Activity Number:    2.2

Activity Title:           Projectile Motion

General Objective:             Gain understanding of the concepts of motion in two-dimensions

Specific Objectives:            (10 sessions)

1.1.                        Describe a projectile

1.2.                        Explain the motion of a projectile using coordinates

1.3.                        Differentiate semi-parabolic projectile from parabolic projectiles

1.4.                        Solve projectile problems using the principles of free-fall and the kinematical equations.

1.5.                        Cite applications of projectile motion.

Description:            

            This activity uses a demonstration-experiment on how projectile motion happens based on their understanding of acceleration and constant speed. Further explanation is given on motions in the text of the workbook to clarify further its meaning.

DIRECTION: Read the discussion, examples and fill-out the blanks on the tables given.

           

            Projectile

           

DEFINITION 2.2.1 missile or shell: an object that can be fired or launched by force.

Projectile Motion

           

           

DEFINITION 2.2.2 the motion or behavior exhibited by a projectile

            What objects exhibit projectile motion?

Projectile motion is exhibited by a body as it is propelled along its path. Below are illustrative examples of the motion of a projectile.

Anybody who has seen a ball shot in a basket in basketball games, or a football kicked such that it bounces up has seen a projectile in motion. All these motions are referred to as rectilinear motion or motion-in-two-dimensions. This motion happens to follow a parabolic trajectory. Due to the nature of forces acting on the object as it travels, the parabolic motion is the normal path of its travel. Examples of such motions besides thrown objects are shown below.

Fig. 2.2.1 A welder cuts holes through a heavy metal construction beam with a hot torch. The sparks generated in the process follow parabolic paths. Parabolic paths are paths of projectiles

Fig. 2.2.2 Pyroclastic eruptions of volcanoes are streams of lava projected upwards. The lava trajectory follows a parabolic path characteristic of a projectile.

 

Description of Motion

The motion of a projectile is best described in two separate equations. When a projectile is thrown, it must have two things in it: An initial velocity v_i and an angle of projection θ_i. A third requirement is time, in order to fully plot the trajectory of the projectile.

Fig. 2.2.3

These are as follows:

Motion along x-axis (along the ground)

                                          where                     EQ. 2.2.1

Motion along y-axis (upward and downward motion)

                                    where                     EQ. 2.2.2

            Q1: Determine the x and y positions of a stone thrown 30˚ above the ground with an initial speed of 60 m/s.  Use a separate sheet of bond paper to solve this problem.

Combining the motion of the two equations by eliminating the time t we come up with this typical x and y equation.

                       A typical parabolic equation   EQ. 2.2.3

            Q2. Derive the Equation 2.2.3 from 2.2.1 and 2.2.2. Use a separate sheet of paper to derive this.

           

            Q3: A ball is thrown in such a way that its initial vertical and horizontal components of velocity are 40 m/s and 20 m/s, respectively. Estimate the total time of flight and the distance the ball is from its starting point when it lands. Use a separate sheet of paper to solve this.

            Determining the Maximum Height and Maximum Range

                                   

Figure 2.2.4

A projectile fired from the origin at t_i = 0 with an initial velocity v_i. The maximum height of the projectile is h, and the horizontal range is R. At A, the peak of the trajectory, the particle has coordinates (R/2, h).

To obtain the maximum height of the projectile shown in Figure 2.2.4, we use this equation:

                        Maximum height equation     EQ. 2.2.4

            To obtain the maximum range (maximum horizontal displacement) of the projectile shown in Figure 2.2.4, we use this equation:

                      Maximum range equation      EQ. 2.2.5

            Q4: Derive the Equation 2.2.4 and 2.2.5 from Equation 2.2.3. Use a separate sheet to show this derivation.

            Q5: A long-jumper leaves the ground at an angle of 20.0° above the horizontal and at a speed of 11.0 m/s. (a) How far does he jump in the horizontal direction? (Assume his motion is equivalent to that of a particle.) (b) What is the maximum height reached? Solve this on a separate sheet of paper.

Figure 2.2.5

A projectile fired from the origin with an initial speed of 50 m/s at various angles of projection.

Note that complementary values of θ_i result in the same value of x (range of the projectile).

Q6: Prove by calculation that the maximum range of an object thrown at 50 m/s is 45˚. Compare its ranges with that of 75˚, 60˚, 45˚, 30˚, 15˚. Solve this on a separate sheet of paper.

                                                                        Figure 2.2.6

           

Q7: A stone is thrown from the top of a building upward at an angle of 30.0° to the horizontal and with an initial speed of 20.0 m/s, as shown in Figure 2.2.6. If the height of the building is 45.0 m, (a) how long is it before the stone hits the ground? Solve on a separate sheet of paper.