where the initial x and y components of the velocity of the projectile are
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1 EXPERIMENT 2: PROJECTILE MOTION OBJECTIVE: Study of a projectile motion on an inclined plane. THEORY PROJECTILE MOTION Projectile motion of an object is simple to analyze if we make two assumptions: (1) the freefall acceleration is constant over the range of motion and is directed downward, and (2) the effect of air resistance is negligible. With these assumptions, we find that the path of a projectile, which we call its trajectory, is always a parabola as shown in Active Figure 2.1. We use these assumptions throughout this experiment. Figure 2.1: The parabolic path of a projectile that leaves the origin with a velocity. The velocity vector changes with time in both magnitude and direction. This change is the result of acceleration in the negative y direction. The expression for the position vector of the projectile as a function of time, with its acceleration being that due to gravity,. (2.1) where the initial x and y components of the velocity of the projectile are (2.2)
2 For a projectile launched from the origin, so that. The final position of a particle can be considered to be the superposition of its initial position ; the term, which is its displacement if no acceleration were present; and the term that arises from its acceleration due to gravity. In other words, if there were no gravitational acceleration, the particle would continue to move along a straight path in the direction of. Therefore, the vertical distance through which the particle falls off the straight-line path is the same distance that an object dropped from rest would fall during the same time interval. Two-dimensional motion with constant acceleration can be analyzed as a combination of two independent motions in the and directions, with accelerations and. Projectile motion can also be handled in this way, with zero acceleration in the x direction and a constant acceleration in the direction,. Therefore, when analyzing projectile motion, model it to be the superposition of two motions: (1) motion of a particle under constant velocity in the horizontal direction and (2) motion of a particle under constant acceleration (free fall) in the vertical direction. The horizontal and vertical components of a projectile s motion are completely independent of each other and can be handled separately, with time as the common variable for both components. HORIZONTAL RANGE AND MAXIMUM HEIGHT OF A PROJECTILE Let us assume a projectile is launched from the origin at with a positive component as shown in Figure 2.2 and returns to the same horizontal level. Figure 2.2: A projectile launched over a flat surface from the origin at with an initial velocity. The maximum height of the projectile is, and the horizontal range is. At, the peak of the trajectory, the particle has coordinates ( ).
3 Two points in this motion are especially interesting to analyze: the peak point A, which has Cartesian coordinates ( ), and the point B, which has coordinates ( ). The distance is called the horizontal range of the projectile, and the distance is its maximum height. Let us find and mathematically in terms of and. We can determine by noting that at the peak. Therefore: (2.3) Substituting this expression for into the y component of Equation 2.1 and replacing with, we obtain an expression for in terms of the magnitude and direction of the initial velocity vector: (2.4) The range is the horizontal position of the projectile at a time that is twice the time at which it reaches its peak, that is, at time. Using the x component of Equation 2.1, noting that, and setting at, we find that Using the identity, we can write R in the more compact form (2.5)
4 INCLINED PLANE The external forces exerted on a block lying on a frictionless incline plane is shown in Figure 2.3. Figure 2.3: The external forces exerted on a block lying on a frictionless incline plane. The net force acting on the block in direction, according to the Newton s law is: Hence the net acceleration of the block in direction is: (2.6) When the projectile motion is studies on an inclined plane, the gravitational acceleration the Equations 2.1 to 2.5 must be replaced by. That is: in (2.7) (2.8) (2.9) THE EXPERIMENTAL PROCEDURE Do not touch the metal parts of the table during the measurements with spark. Never switch on the spark if one or two pucks are out of table. Do not keep one of the pucks at your hand when spark is on. Otherwise you may get an harmless but disturbing electric shock.
5 Incline the air table as shown in Figure 2.4 with the help of a wooden block. Figure 2.4: Schematic drawing of an inclined air table Measure the lengths h 1, h 2 and L of the inclined table. Write the results to the table below. Place the carbon and measurement paper to the air table. Place the air puck launcher to the BOTTOM RIGHT corner of the air table with an angle you desire. Place the rubber band to the second position on the launcher. Place one of the air puck to the launcher and fix the other puck to the top corner with the help of a plastic fixer in such a way that it doesn t block the motion of the other. (Figure 2.5). Figure 2.5: Projectile method. Run the air pump and try projectile motion few times until you get enough experience. Switch on the spark timer and choose a value (40,60 or 80 ms). Press the red spark button in synchronization with the projectile motion that you have started with the launcher. Keep the button pressed until the end of the motion. Switch off the air pump and spark timer to see your measurement results. The laboratory supervisior should confirm your chosen measurement when you are sure about your measurement.
6 Draw the x and y coordinate projections as shown in Figure 2.6. Take the second measurement point after starting point since the starting point can be problematic and place the corner of your coordinate axes to this point. Measure and values and record it to the table below. Figure 2.6: Measurement of the projectile motion data. Determine and flight times by counting the points on projectile motion. Measure the lengths in order to determine the starting ) and the end ) angles as shown in figure 2.7. Use more points as much as possible in such a way that the linearity of the line is not violated. Extend the line in order to increase the precision of the measurement. Figure 2.7: Angle measurement of the initial and final velocities.
7 CALCULATIONS 1. Determine the angles and with the help of equations given below. 2. Determine the inclination angle of the air table by using the equation below. (2.10) (2.11) 3. Use gsinα instead of g in all projectile motion equations since the projectile motion is carried out with an inclined air table. Figure 2.8: A sample projectile motion data view. 4. Take the average of the equal distances on x axes between the points. 5. Calculate the x component of the initial velocity by using the equation below: (2.11) where t is the multiplication of the number of points N on paper. This gives us the flight time experimentally. 6. Calculate the initial velocity experimentally by using the equation. Calculate also the end velocity using the same equation. 7. Can and angles be different from each other? Make comments on the corresponding reasons and the results. 8. Calculate the experimental value of the time to reach maximum height by using number of points N: (2.12) 9. Calculate the theoretical value of the time to reach maximum height by using the equation (2.7). 10. Calculate the time t = 2t A to reach the range R experimentally. 11. Measure the experimental value of the range R exp by using a ruler. Calculate the same value theoretically by using the Equation 2.9.
8 12. Measure the experimental value of the maximum height by using a ruler. Calculate the same value theoretically by using the Equation Write your results to the corresponding table below and make the error calculation for each result you obtained by using; (2.14) MEASUREMENTS TABLE Time (t) (ms) Time to reach maximum height ( ) (s) Flight time t(s) Range (R) Maximum height (h max ) Measured Values RESULTS TABLE ( 0 ) ( 0 ) α ( 0 ) (cm/s) (cm/s) Time to reach maximum height ( ) (s) Flight time t(s) Range (R) Maximum height (h max ) % Error
9 COMMENTS QUESTIONS 1. Which parameter remains constant for an object making a projectile motion? a. Magnitude of its velocity, b. Acceleration, c. Horizontal projection of its velocity, d. Vertical projection of its velocity. 2. A ball hit by a soccer player has an initial velocity of 30 m/s and it makes 37 o angle with horizontal axes. a. Write the equation of motion of the ball. b. What is the position and velocity components of the ball at t=1s? c. How long it takes to reach the maximum height? d. What is the maximum height that ball reaches? e. How far the ball goes before hitting the ground? 3. A ball is thrown away from the roof of a 33 m high building, with an angle of 53 o with horizontal and with an initial speed of 5 m/s. At ground, a children starts running horizontally at the same time with a constant acceleration of. a. Write the equation of motion of the ball by choosing a coordinate system. b. What is the flight time of the ball? c. How far away from the building the ball goes? d. What should be the acceleration of the children to be able to catch the ball before hitting the ground?
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