Center for Technology and Teacher Education: Mathematics Activities

Two groups of students drive from Clemson, SC to Charlottesville, VA to attend a UVA-Clemson football game. The first group leaves Clemson at 10 AM and travels at a uniform speed of 45 miles per hour. The second group leaves at 11:30 AM and travels at a speed of 60 miles per hour. At what time and distance from Clemson will second group catch the first group?

(a) Solve this “problem” algebraically.

(b) Use parametric equations to simulate the travel of these cars on a graphing calculator.

(c) Solve this problem by graphing these equations and tracing the graphs or generating a table.

Hint: Fix one graph at y = 1 and the other at y = 2. Hence x1(t) = 45t, y1(t) = 1 and x2(t) = 60(t-1.5), and y2(t) = 2. Use simultaneous graphing.

Note: Attention must be paid to determining "good" windows, to not only allow "nice" tracing coordinates, but also to display the traveling cars at observable speeds (i.e., use a small t-step).

(d) Use both algebraic and graphing calculator methods to solve the following:

A motorist leaves Charlottesville at 4 am and drives towards Manhattan at a uniform speed of 65 miles per hour (Notice that he passes DC before rush hour). At 6 am, his sister leaves Manhattan and drives towards Charlottesville at a uniform speed of 75 miles per hour. Where, and at what time will they meet? (Assume Manhattan is 360 miles from Charlottesville and that the George Washington Bridge is not backed up).

(e) Comment on the different methods to solve this type of problem and the strengths and weaknesses of each.

(a) Suppose, under conditions of no gravity and no air resistance, that a spherical projectile is launched vertically by an instantaneous force which gives it a constant velocity of 98 m/s. Algebraically, determine the height above the surface of the earth the projectile will be at 10 seconds and 20 seconds.

(b) Simulate the path of this rocket using parametric equations, and observe and trace the path.

(c) By looking at the graph, predict the derivative of this function at several points, and then use your calculator to compute the numerical derivative.

(d) Comment on the differences between the algebraic and graphical approaches to solving this problem.

(e) Suppose now that the projectile was not launched, but instead fell off a very high cliff straight downward due to the force of gravity. Determine how far the projectile would fall after 10 seconds and after 20 seconds.

(f) Simulate the path of this falling projectile using parametric equations, observe and trace the path.

Hint: Use the equation relating distance traveled to acceleration, d=.5at².
Suggestion: Fix this graph at x=2. Hence, x₂(t) = 2, y₂(t) = -.5(9.8)t². Suggestion: Use plot mode when graphing and use a window that will highlight the acceleration.

(g) Use your observations of the distance fallen at successive points in time to estimate the value of the derivative of this function at several points, and use your calculator to compute the numerical derivative to test your predictions. Relate these to the function.

(h) Observe the change in distance fallen between successive seconds, and relate your observations to acceleration, and the second derivative of the function.

(i) Now suppose that the projectile was launched vertically as in (a) above, but was also under the influence of gravity. Observe the parametric graphs of the upward projectile in (b) and the downward projectile in (f), and predict where a projectile under both the same upward force and gravity would be after 10 seconds and after 20 seconds.

(j) Determine parametric equations for the path of this projectile, and simulate the path on your calculator.

(k) Simulate the path using parametric equations so that the upward portion and downward portion of the path do not coincide on the screen.

Hint: Adding some small linear function to the x-coordinate (e.g. x₃(t) = 3+(t/320)) moves the downward portion of the graph over 1 pixel. This function will depend on your window.

(l) Predict the derivative of this function at several points, and use your calculator to compute the numerical derivative.

(m) Determine a Cartesian equation for the path of this projectile and use it to generate a graph.

(n) Trace each graph, and comment on differences between the Cartesian and the parametric graphs.

(a) Suppose, under conditions of no air resistance, that three similar projectiles are launched at angles of p/6, p/4, and p/3 radians (or 30, 45, 60 degrees) with an initial velocity of 36 m/s. Predict which projectile will go the highest, which will travel the furthest, and which will take the longest time to hit the ground. Estimate how long it would take each to hit the ground.

(b) Simulate the paths of each of these projectiles with parametric graphs.

(c) Observe and trace the various paths. Comment on your observations.

(d) Estimate the x and y derivatives at the upward, maximum, and downward portions of each graph.

(e) Use your calculator to determine numerical derivatives at these portions of the graphs. Comment on your observations.

In reality, projectiles are under the influence of a number of forces. In addition to both the instantaneous force applied to launch a projectile and gravity, a force due to air resistance tends to slow projectiles down. The surrounding air produces a “drag” force which operates against the motion of a projectile. This drag force depends on a number of variables, including the size, shape, and orientation of the projectile, the density of air, and the square of the velocity of the projectile. This drag force can be expressed by the drag equation:

Drag = C_d*V²*S, where C is an experimentally determined coefficient which depends on several factors, V is the velocity of the projectile, and S is the surface area of the projectile normal to its direction.