Laboratory #6: Measuring g using Atwood's machine



Laboratory #5: The Muzzle Velocity & Impulse of a Nerf Gun

Purpose:

1. To study the motion of a freely falling object.

2. To determine the maximum height and muzzle velocity of a projectile shot from a Nerf Cannon

3. To determine the impulse delivered to a projectile shot from a Nerf Cannon

Introduction: General Otto von Nerf has developed the ultimate weapon of mass destruction, the Nerf Blaster cannon. You have been assigned the task of determining the vital operating characteristics of this lethal device, namely the muzzle velocity and maximum height of a launched projectile.

Using the laws of kinematics you will analyze this device. First, the change in velocity/speed of an object under constant acceleration can be determined by the relationship:

Δv = vfinal - vinitial = at

Since the acceleration on the object while it is in flight is strictly due to gravity (neglecting air resistance), the equation can be re-written as:

Δv = vfinal - vinitial = - gt

where g is 10 m/s2 and t is the hang time (or thang) of the launched ball. In this equation, vinitial is the muzzle velocity (vmuzzle) of a projectile launched from the cannon. It should be noted that since air resistance will not be considered, it can be assumed that the magnitude of the speed of the object at the end of its flight (i.e. the bottom of its trajectory) is the same as the muzzle velocity. Also, as the projectile comes down it is moving in the opposite direction as when it was shot up, thus we can say:

vfinal = - vinitial = - vmuzzle

Thus, we can re-write our motion equation for the muzzle velocity as:

Δv = - vmuzzle - vmuzzle = - gt

or,

Equation 1: vmuzzle = ½ . gt

This equation will be utilized to determine the muzzle velocity of a projectile launched from the Nerf Blaster.

To determine the maximum height of a launched projectile, a separate kinematic equation will be utilized. The distance traveled by an object under constant acceleration is given by the relationship:

Equation 2: d = ½ gt2

This equation relates the distance traveled by an object projected from the cannon to the time the object is in flight. Since we are interested in the height of the projectile’s trajectory, the time, t, in this equation refers to the time elapsed for the ball to reach its highest point. In other words, t is half of the hang time of the launched ball, or

t = ½thang

We will use this relationship to determine the maximum height of a projectile shot vertically from a Nerf cannon.

Preliminary Questions:

1) For a freely falling object, what is the acceleration (ignore air resistance)?

2) Consider a ball shot directly upward into the air (free fall) from ground level.

a) How does the time on the way up compare to the time on the way down?

b) How does the speed of the ball immediately after being shot compare with the speed when the ball returns to the same height as it was shot (on the way down)?

c) How do the corresponding velocities in (b) compare?

3) How is the change in velocity of a ball related to its acceleration and elapsed time?

4) How is the distance a ball travels related to its acceleration and elapsed time?

Apparatus:

• One meter stick

• a Nerf cannon and ball

• stop watch.

{Optional}

• ULI & Photogate sensor

• Computer w/LoggerPro Software

Procedure:

Place the butt of the nerf cannon against the ground and point barrel as vertical as possible. Push the forward grip all the way forward then pull it back toward the back grip to launch a ball. Use a stop watch to measure the passage of time as the ball shoots from the cannon. You want to record the time for the ball to fly upward and return to the initial height of the gun barrel Repeat 4 times. Enter your data in the table below. Using a calculator, determine the average hang time Warning: Do not shoot your classmates with the Nerf Cannon.

Using the equations above estimate the average muzzle velocity and the average height of the trajectory.

|Trial # |Hang Time (thang) | |

|1 | |{Estimated Values} |

|2 | |Avg. Muzzle Velocity (vmuzzle) |

|3 | | |

|4 | |Avg. Height (d) |

| tavg = | | |

Verification:

A) Muzzle Velocity

{Choose One of the Two Options}

1) Set up a photogate across the barrel of the Nerf Cannon (instructor will provide details). The experiment file for this method is “PHY101-Exp04”. Directly measure the muzzle velocity for 4 launched balls. Calculate the average muzzle velocity for these measurements. How does the directly measured average muzzle velocity compare with the estimated value determined above? Determine the experimental uncertainty (%). Can you think of a better method to measure the muzzle velocity? If so, try it!

|Trial # |vmuzzle |avg. vmuzzle |avg. vmuzzle |Experimental |

| |(measured) |(measured) |(estimated) |Uncertainty (%) |

| | | |(e.g. from previous table) | |

| | | | | |

| | | | | |

| | | | | |

| | | | | |

2) Point the Nerf Cannon horizontally. When you shoot the cannon, measure both the time it takes for the projectile to fall to the ground and measure the distance the projectile travels horizontally during the fall. The horizontal velocity should not change (neglect air resistance) until the projectile hits the ground. Use this information to calculate the muzzle velocity.

|Trial # |Fall Time |Horizontal |vmuzzle |avg. vmuzzle |avg. vmuzzle |Experimental |

| | |Distance |(measured) |(measured) |(estimated) |Uncertainty (%) |

| | | | | |(e.g. from previous | |

| | | | | |table) | |

| | | | | | | |

| | | | | | | |

| | |. | | | | |

| | | | | | | |

| | | | | | | |

B) Trajectory

Using a meter stick, directly measure the maximum height for 4 launched balls. Calculate the average height for these measurements. How does the directly measured average height compare with the estimated value determined above? Determine the experimental uncertainty (%).

|Trial # |h |avg. h |avg. h |Experimental |

| |(measured) |(measured) |(estimated) |Uncertainty (%) |

| | | |(e.g. from previous table) | |

| | | | | |

| | | | | |

| | | | | |

| | | | | |

Summary Questions:

1) Do your calculated (estimated) results for muzzle velocity and height of projectile flight agree with each other? Why? Why not?

2) Do your results support the assumption that air resistance is negligible during the projectiles flight? Explain.

3) Examine the individual measurement values for each set of data you acquired. How close are the values to each other? Should the values be close? Are they? Are they not? Why?

4) Determine the average impulse delivered by the Nerf cannon to the projectile. {Hint: You will need to measure the mass of the Nerf balls}

Setting-up the Photogates:

1) run Logger Pro program

2) set-up Logger Pro

1. select File(New {create a new experiment}

2. add sensor ( choose photogate from menu

3. data collection ( set to “Photogate Timing”

click on “Mode”

select “Gate Timing – One Gate”

measure diameter of the balls (calculate average)

enter ball diameter as “Length of Object”

click on “OK”

3) set-up photogate with ring-stand

4) place barrel of cannon just below photogate (be sure to center barrel in gate)

5) fire gun

6) repeat several times to perfect your measurement technique

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