Matter & Interactions 2nd ed. Practice Problems
Aaron Titus | High Point University
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1320001     The net force on a dart in a spring-loaded dart gun.     1320001
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Question

 

A spring-loaded dart gun is used to shoot a 5-gram dart straight upward. Its spring has a stiffness 50 N/m, and its relaxed length is 5.0 cm. The spring is compressed 4.0 cm from its relaxed length. What is the velocity of the dart when it leaves the spring (i.e. at the instant the spring returns to its relaxed length)? Apply the Momentum Principle iteratively in order to answer the question.

 



Solution

 

Use an iterative approach to answering the question. In other words, write a program that calculates the net force on the dart, its momentum, and its position in small time steps (i.e. small time intervals). The easiest way to do this is to write a program, perhaps in Python.

In an iterative loop, the program should do the following for each object in the simulation:

(1) Calculate the (vector) forces acting on the object and sum the forces to calculate the net force on the object.

(2) Update the momentum of the object: .

(3) Update the position of the object , where the average velocity can be approximated as . (Note: there are more accurate ways to approximate the average velocity.)

(4) Update the clock reading, .

The loop repeats, performing each of the calculations above for each time step, .

You can use a condition in the while loop or an if statement to stop the loop and print the position and momentum of the dart. Or, you can simply print the position and momentum of the dart in an infinite loop and after enough iterations, manually stop the simulation and examine the data in order to answer the questions.

Begin by drawing a picture of the situation, with a coordinate system. This is important for determining the initial position of the dart, for example.


Figure: A dart as it is launched from a dart gun.

While we can certainly develop realistic graphics in the simulation, let's begin with a simple simulation that does the correct calculations, but without fancy graphics. Thus, the simulation will model the dart as if it's a sphere and will only show the dart and the base of the barrel of the dart gun. In addition, we'll define y = 0 to be the center of the ball when it leaves the dart gun.


Figure: A simple model for the dart gun.

The initial position and initial velocity of the dart is:

The initial amount that the spring is stretched is:

Because of our choice of coordinate system, it is convenient that the distance stretched is equal to the negative of the dart's y position.

This allows us to write the force by the spring on the dart as

But this holds true for any position of the dart/spring system, not just the initial position. Thus, while the dart is in contact with the spring,

The above relationship between force by the spring on the dart and the position of the dart is only valid because of our choice of the coordinate system. If we did not choose our coordinate system so conveniently, then you would have to explicitly calculate the distance the spring is compressed at any instant during the motion.

Now, draw a free-body diagram and write the net force on the dart.


Figure: A free-body diagram for the dart.

The net force on the dart is

Now that we have the expression for the net force on the dart and now that we have its initial conditions, we can write the simulation. Here is an example program in Python to simulate the motion of the dart. This program does not show the spring.

#Simulation of a dart launched vertically from a dart gun

from visual import *

#set the scene so vpython won't autoscale as we watch the animation scene.autoscale=0 scene.range=0.1

#create the dart and the base of the dart gun; the dart is shown as a sphere dart=sphere(radius=0.01, pos=(0,-0.04,0), color=color.yellow) base=box(pos=(0,-0.06,0), size=(0.015,0.002,0.015), color=color.white)

#define constants k=50.0 #N/m g=vector(0,-9.8,0) #m/s^2 m=0.005 #kg

#define initial conditions dart.pos = vector(0,-0.04,0) dart.v = vector(0,0,0) dart.p = m*dart.v

t = 0 dt = 0.0005

#print a header for data print "time (s)", "\t", "position (m)", "\t", "velocity (m/s)"

#stop the loop when the y-position of the dart reaches zero while dart.pos.y < 0 :

rate(10)

Fspring = -k*dart.pos Fgrav = -m*g Fnet = Fspring + Fgrav

dart.p = dart.p + Fnet*dt dart.v = dart.p/m dart.pos = dart.pos + dart.v*dt

t = t + dt

print t, "\t", dart.pos, "\t", dart.v

The last four data points printed are:

 

time (s) position (m) velocity (m/s)
0.014
0.0145
0.015
0.0155

Table: Position of the dart as it is launched, before it leaves the spring.

 

The dart leaves the spring during the interval between t = 0.0150 s and t = 0.0155 s. Its y-velocity when leaving the spring is about 4.1 m/s, rounded to two significant figures.

It's important to solve this problem iteratively because the net force on the dart is NOT constant. This is apparent by graphing the y-momentum as a function of time. It is not linear. If the y-component of the net force on the dart were constant, then the y-momentum vs. time graph would be linear.


Figure: A graph of y-momentum vs. time for the dart. Note that it is not linear.