Assignment 2

Name:
Student #:

1 Hitting a target

A projectile of mass 2.30 kg is fired from a height of 20.0m above a level field, 15.0m away horizontally from a target.  If the muzzle speed is 10.8 m/s, at what angle(s) should the projectile be fired to hit the target if the projectile feels a drag force of (Subscript[F, d])=-0.200 v, where v is the velocity of the projectile?  Express your answer in degrees.  Use g=9.81 m/(s^2).

There are a variety of ways of answering this question.  However, feel free to follow these steps.

step 1

Set up and solve Newton’s 2nd law for the x and y positions as functions of time and firing angle, given all the other numerical parameters.  Store your solutions in functions called X[t,θ] and Y[t,θ].

step 2

Having obtained X[t,θ] and Y[t,θ], issue the following command that allows you to visualize the trajectory for value of θ between 0 and π/2 radians (0 to 90 degrees).

Manipulate[ParametricPlot[{X[t,θ],Y[t,θ]},{t,0,4},PlotRange->{{0,25},{0,27}}],{θ,0,π/2}]

This should produce an interactive graphic with a slide bar that allows you to change θ.  Clicking the little “+” symbol allows you to see a numerical value for θ when you work the slide bar.  Play with the result until you find an estimate for θ that results in the target being hit (x = 15 when y = 0).  Call this angle θestimate.  This procedure also confirms that the problem is solvable, e.g., the target is not too far away to be hit in the first place.

step 3

Plot the y position of the projectile as a function of time when it is fired at an angle of θestimate.  Use these plot to find the time at which the projectile hits the ground.  Call this time “testimate”

step 4

Use FindRoot to find precise values of time and θ at which the projectile hits the target.  Store these numbers in variables called thit and θhit.  Confirm that these are reasonable solutions by plotting the trajectory of the projectile from t=0 to thit when it is fired at an angle of θhit.  Convert θhit to radians.

2 Numerically finding the optimal firing angle

Part 1 - a Do loop (no marks)

It is often necessary to perform repetitive calculations, where one variable is varied systematically.  Study the follwing code that produces a table of values representing the funtion y=(x^2) from x=-2 to 3 in small increments

Remove["Global`*"]

xstart=-1;
xstop = 2;
xincrement=0.2;
Do[y[x]=x^2;
Print[x," ",y[x]],{x,xstart,xstop,xincrement}]

(In this Do loop, a series of statements are executed for a sequence of values of x, starting from -1 and increasing progressively by 0.2 until the value of 2 is reached.  In executing these lines, we have produced a function y[x] that is defined only at specific values of x.)

ytab=Table[{x,y[x]},{x,xstart,xstop,xincrement}]

(This line represents the data contained in y[x] as a Table, which we can plot with ListPlot.)

ListPlot[ytab]

Part 2 - a projectile problem (1 mark: see Section 3.6 of online notes)

A projectile of mass 1.13 is shot at a given speed into the air at a certain elevation angle over a level field.  Air resistance gives a drag force of magnitude F = b (v^2) and directed antiparallel to the velocity, i.e. (OverVector[F])= - b (v^2) (OverHat[v ]),  (Subscript[F, x])=-b (Subscript[v, x])(Sqrt[((Subscript[v, x])^2)+((Subscript[v, y])^2)]) and similarly for (Subscript[F, y]).
	Plot the trajectory for firing angle  θ=35.0 °, initial speed v0=12.0 m/s, g=9.81 and b=0.427.
	Find the range
	Plot the trajectory

part 3 - Range as a function of firing angle (9 marks)

Repeat the same kind of calculation as in part 2, except within a Do loop in which the firing angle is varied between 10 and 80 degrees in 1 degree increments.  In the end, you should have an estimate of the optimal firing angle (obtained from a table of values), as well as a plot displaying the results (with axes labels that include units) .

3 A little rocket

A self-propelled projectile is fired into the air at angle of θ=40 degrees with an initially speed of 20 m/s.  The projectile faces a drag force of magnitude 0.3 v, where v is the projectile’s velocity.  The magnitude of the force produced by the little engine on the projectile is given as a function of time by R(t)=50 exp(-t/10), and is directed in the direction of the velocity.  As the projectile burns fuel, its mass decreases.  The mass as a function of time is 0.1 (1+exp[-t/10]).  Assume MKS units and a gravitational acceleration of g=9.8.

Assuming the projectile flies over a level field, find the time of flight and the range.

Plot the trjectory of the projectile.