Investigation of horizontal projectile motion
Research
The main aim of the experiment is to determine the angle at which the horizontal speed is fastest in projectile motion. This will be achieved by launching a ball using a spring gun on the horizontal direction and then repeating this procedure for a number of times at different elevations. The readings are then recorded and analyzed to determine the speed at different angles.
My research question is “for what angle will the horizontal speed be the fastest?” An object in that is in free fall tends to accelerate at the rate of 9.81 m/s2. When carrying out an investigation on the trajectory that is created by the projectile motion, both the horizontal and vertical motion must be considered independent of each other. On the horizontal direction, there is no acceleration and the velocity is constant. The initial velocity in horizontal projectiles is usually 0 m/s2.
In this experiment, a plastic ball was fired from the top of an elevated spring gun at different angles of inclination. The horizontal distance that was travelled by the ball was measured and recorded. The test was carried out for a number of trials and the average values obtained and recorded for further analysis. The initial velocity calculates proves that the values are dependent on the angle of elevation and are fairly constant. Some of the possible sources of errors in this experiment could be zero error and the effect of resistance due to air resistance
Theory
A projectile can be defined as a physical body thrown into space with some initial velocity after which it is allowed to move under the influence of the force of gravity A trajectory is the path that is followed by a projectile. In the horizontal direction, the projectile moves at a constant acceleration at 9.8 m/s2 towards the vertical direction. The speed of the projectile (body) remains constant on the horizontal direction implying that it does not change until the moving body hits its target. The position of the projectile at a particular time is given by the equation:
X=V_x t
For a projectile that is launched at a given angle of inclination, the height and the range of motion can be determined. It is important to point out that whenever an object is projected at an angle that is steep, it tends to spend much more time in the air that it does when launched at a shallow angle. An object that is launched at a shallow angle moves faster on the horizontal direction as compared to that launched at a steep angle. Maximum range is achieved when an object is launched at an angle of 45o.
Apparatus
The materials needed for this experiment include:
Table
Clamp
Plastic ball
Wooden block,
Ruler
Inclinometer
Spring gun
Pendulum
Height gauge.
Method
The equipment to be used for the experiment were arranged on a lab table and the pendulum moved away to prevent it from blocking the firing spring gun. A white paper with a carbon paper on top of it was spread on the floor act as the target where the position that the fired ball would be landing. To ensure that the firing mechanism was functioning properly, the spring gun was test fired and the position of the target marked. The spring gun was fired a number of times for each trial and the readings for the horizontal distance and time taken by the ball on the target recorded. Using a wooden block, the angle of inclination was adjusted and measured using an inclinometer. The gun was fired and the readings taken before adjusting to the next angle. The readings taken were done at 00, 150, 300, 450, 600, 750 and 900.
Figure 1: Diagramatic representation of the experimental set up
Data and analysis
The initial velocity remained constant for each trial. The spring gun was fired at an elevation of 0.5 m. The values obtained are recorded in the table below.
Elevation=0.5 m
g=9.81 m/s2
Trial Angle of inclination Time of flight (s) Measured Horizontal distance (R) (m) Calculated Horizontal distance (R) (m) Velocity of the ball (m/s) Deviation
1 0 2 10 19.62 25 ±0.05
2 15 5 20 122.625 8.69 ±0.05
3 30 8 40 304.32 3.25 ±0.05
4 45 10 60 490.5 1.77 ±0.05
5 60 15 40 1103.625 0.67 ±0.05
6 75 18 20 1589.22 0.216 ±0.05
7 90 21 10 2163.105 0 ±0.05
The height of the ball was calculated using the height and the acceleration due to gravity
h=1/2 gt^2
For the first trial h=1/2 x9.81×2^2=19.62 m .The calculation was done for all the other trials.
To determine the velocity of the ball for each trial Speed=distance/time
Velocity along the horizontal component=S_x Cosθ
For trial 1, the velocity is S_x Cosθ=25Cos 0=25 m/s. The calculation was done for all the other trials.
The horizontal distance (R) travelled by the ball was measured using a meter rule and then compared against the expected horizontal distance.
R=(v_0 cosθ)t
For trial 1, the expected horizontal distance was computed as
R=(v_0 cosθ)t=(25Cos0)2=50 m
The values of the expected values of R were computed for all the other trials.
The uncertainty incurred along the distance travelled by the ball was determined by considering the deviation and the average value of the distance.
Uncertainity=∆R/R x100=0.05/27.86 x100=0.179%
The results obtained indicate that the optimum angle for the velocity is 00 and 450 for the range. Using the angular values obtained from the experiment, the initial velocity was computed using the values of the horizontal velocity.
d_x=(v_i^2 sin2θ)/g
The equation used to solve for the initial velocity was
v_i=(d_x g)/sin2θ
The values for the initial velocity are recorded in the table below
Angle of inclination Initial velocity (m/s)
0 0
15 19.81
30 21.28
45 9.90
60 13.04
75 19.81
90 0
The values above show a constant initial velocity. There is very little difference between the values and the average value of the extreme velocity.
Figure 1: Graph of velocity versus inclination angle
Figure 2: Graph of measured angle versus angle of inclination
Discussion
From the results obtained, there seems to be a peculiar relationship between the angle of inclination and the velocity of the ball. The ball is fastest at a shallow angle of 00 and slowest at a wide angle of 900. Velocity is a function of speed and direction. When the ball was launched at an angle of 00, there was very little deceleration as a result of the effect of gravity. As a result, the ball moved faster. As the angle of inclination increased, the effect of the force of gravity increased thus resulting into a rise in the rate of deceleration. At 450, the maximum range was achieved. The graph of the angle of inclination versus the velocity of the ball indicates a reduction in the velocity as the angle increases. On the other hand, the graph of range versus angle indicates a line of symmetry at an angle of 450.
The calculated values of the range and the velocity confirm the validity of the equations of horizontal motion used in the experiment. The concepts of horizontally projected motion have found wide use in science. Rocket missiles are launched at particular angles to ensure that they are able to hit their targets. Sportsmen such as footballers and athletes use this concept to achieve maximum range when kicking the ball and when throwing javelins and discus. This concept is also used in training archers and shooters.
Sources of error
The experimental data obtained shows some discrepancy as compared to the expected values. Another indicator of error is also evident in the deviation of ±0.05 in the measurement of the horizontal distances. Some of the possible sources of error in this experiment include zero error on the measuring ruler and air resistance as the ball moves across the air. The resistance experienced when the ball is in motion can slow down the ball therefore resulting in a reduction in the measured values in air. They would appear to be lesser than the expected values. The firing gun was also raised above the ground and the influence of this height is not accounted for in the calculations.
Conclusion
From the lab experiment, the various concepts of horizontal projectile motion were determined. It has been proved that both the horizontal and the vertical motions of the object are not dependent on each other. The only common factor that they share is time while allows for the correlation of the two vector motions. When a projectile is launched at an angle, the speed decreases with an increase in the angle of inclination. The range of the object however increases up to an angle of 450 and then decreases to 0 at an angle of 900.
References
