To Study the Variation in Range of a Projectile with Angle of Projection
Objective
To experimentally verify the variation in the range of a projectile with different angles of projection and to determine the angle that gives maximum range.
Theoretical Background
When an object is projected at an angle to the horizontal, it follows a parabolic path under the influence of gravity. This motion is known as projectile motion and is a combination of horizontal and vertical motion.
For a projectile launched with initial velocity $u$ at an angle $\theta$ with the horizontal:
Horizontal component of velocity: $u_x = u \cos \theta$
Vertical component of velocity: $u_y = u \sin \theta$
Time of flight: $T = \frac{2u \sin \theta}{g}$
Maximum height: $H = \frac{u^2 \sin^2 \theta}{2g}$
Range: $R = \frac{u^2 \sin 2\theta}{g}$
The range is maximum when $\sin 2\theta = 1$, which occurs when $\theta = 45°$
The range R of a projectile is given by:
$$R = \frac{u^2 \sin 2\theta}{g}$$
where $u$ is the initial velocity, $\theta$ is the angle of projection, and $g$ is the acceleration due to gravity.
Materials Required
- Projectile launcher or spring gun
- Meter stick or measuring tape
- Protractor for measuring angles
- Stopwatch (optional)
- Carbon paper or chalk powder (to mark landing positions)
- Graph paper
- Calculator
Experimental Setup
Procedure Setup:
- Place the projectile launcher on a level surface at one end of the laboratory.
- Ensure the launcher has a mechanism to adjust and measure the angle of projection.
- Mark the horizontal distance with tape or chalk marks.
- Set up the carbon paper or chalk powder at the landing area to mark the exact point of impact.
Experimental Procedure
- Set the launcher to the desired angle (start with 15° and increase in steps of 5° or 10° up to 75°).
- Load the projectile in the launcher, ensuring the same initial velocity for all trials.
- Release the projectile and mark its landing position.
- Measure the horizontal distance from the launcher to the landing point (the range).
- Repeat steps 1-4 for each angle, performing at least 3 trials per angle for reliability.
- Calculate the average range for each angle.
- Record your observations in the data table.
Data Collection
| Angle of Projection (θ°) | Trial 1 Range (cm) | Trial 2 Range (cm) | Trial 3 Range (cm) | Average Range (cm) | sin 2θ | Theoretical Range (cm) |
|---|---|---|---|---|---|---|
| 15° | 0.500 | |||||
| 20° | 0.643 | |||||
| 30° | 0.866 | |||||
| 45° | 1.000 | |||||
| 60° | 0.866 | |||||
| 70° | 0.643 | |||||
| 75° | 0.500 |
Calculations
Calculating Initial Velocity
If we know the maximum range $R_{max}$ occurs at $\theta = 45°$, we can calculate the initial velocity $u$ using:
$$u = \sqrt{\frac{R_{max} \times g}{1}}$$
Calculating Theoretical Range
Once we know the initial velocity, we can calculate the theoretical range for each angle using:
$$R_{theoretical} = \frac{u^2 \sin 2\theta}{g}$$
Calculating Percentage Error
$$\text{Percentage Error} = \frac{|R_{theoretical} - R_{experimental}|}{R_{theoretical}} \times 100\%$$
Graphical Analysis
Plot the following graphs:
- Range (R) vs. Angle of projection (θ)
- Range (R) vs. sin 2θ
The Range (R) vs. Angle (θ) graph should form a parabolic curve with the maximum value at 45°.
The Range (R) vs. sin 2θ graph should be approximately linear according to the equation:
$$R = \frac{u^2}{g} \sin 2\theta$$
The slope of this line gives the value of $\frac{u^2}{g}$, which can be used to calculate the initial velocity if g is known.
Analysis Questions
- At what angle did you observe the maximum range? Does it match the theoretical value of 45°?
- How does the experimental relationship between range and angle of projection compare to the theoretical relationship?
- Calculate the initial velocity of your projectile using the maximum range measured.
- Explain any discrepancies between theoretical and experimental values.
- If two angles give the same range, what is the relationship between them? Verify this from your data.
- How would the range vs. angle graph change if this experiment was performed on the moon?
Possible Sources of Error
- Air resistance affecting the projectile motion
- Inconsistent initial velocity between trials
- Errors in angle measurement
- Uneven launching surface
- Human reaction time for measurements
- Parallax errors in reading measurements
Conclusions
Based on your experimental results, write a conclusion addressing the following points:
- Verification of the angle for maximum range
- Relationship between range and sin 2θ
- Validity of the equation R = u²sin 2θ/g
- Discussion of experimental limitations