Enrichment, Extension, and Application

Question 1

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Exercises

  1. Riding, running, and walking...speed that counts!
    1. Michael goes for a ride on his bicycle. He rides at \( 30 \) km/h for the first half of the distance, and \( 20 \) km/h for the second half. What is his average speed for the whole trip?
    2. Nana and her granddaughter Sandy leave the house at the same time for their daily exercise. Nana takes a leisurely walk around the neighbourhood while Sandy runs several times around the same route, overtaking Nana twice during her run, and then arriving home at the same time.

      Next day, Sandy decides to run in the opposite direction to Nana. If they both move at the same (constant) speeds as before, how many times will they meet?

  2. Objects in motion.
    1. An object travelling in a straight line has position defined by \[ s(t) = 2 t^3 - 12 t^2, t \geq 0 \] where \( s \) is measured in metres and \( t \) in seconds.
      1. Find the velocity and acceleration of the object at \( t = 1 \) seconds.
      2. Does the object start from rest at \( t=0 \)? Explain.
      3. What is its position when the object stops? Describe the subsequent motion of the object.
      4. Determine whether the speed of the object is increasing or decreasing at \( t=3 \) seconds. Explain your answer by describing the object's motion at that time.
    2. Earth with trajectory of a rocket shown as an arrowed line
      NASA. [Photo]. Retrieved from http://www.nasa.gov/images/content/
      115334main_image_feature_329_ys_full.jpg
      Modified for educational use.

      Two satellites travel directly toward one another along a semi-elliptical path above the earth, with a maximum altitude (the semi-major axis) of about \( 1200 \) km. One is moving at a speed of \( 2000 \) km/h and the other at \( 1000 \) km/h. If they are initially \( 5000 \) km apart (measured along their path), how many kilometres apart are they one minute before they bypass one another?

  3. Docking gently.

    With its motor lifted at \( t=0 \), a boat glides towards a dock. Its distance (in metres) from the dock is given by \[ s(t) = \frac{s_0}{2} \left( \frac{6-t}{3+t} \right) ~ \] for \( t \geq 0 \), where \( s_0 \) is a positive constant.

    1. Find the boat's velocity \( v(t) \) and explain its sign.
    2. If the driver wishes to reach the dock with a speed of \( 0.5 \) metres per second, how far from the dock should the driver lift the motor?
  4. Definitely not a beach ball! Sphere with radius a, ellipsoid with great circle of radius r in horizontal plane and height h

    An exercise ball is initially spherical with radius \( a \) but changes shape when in use.

    Suppose for \( t \gt 0 \) the new shape is ellipsoidal (as shown at the right), and hence has volume \( V_e = \frac{2}{3}\pi r^2 h \).

    1. Assuming the air inside the ball is not compressed, explain why the height \( h(t) \) must decrease as the radius \( r(t) \) increases.
    2. Show that at the instant when \( \dfrac{dh}{dt} = -2\dfrac{dr}{dt} \), the radius is \( r = \left( 2^{\frac{1}{3}} \right) a \).
  5. Graphs galore!
    1. At \( t = 0 \), a car going due north at \( 120 \) km/h on a straight highway passes a motorcycle policeman going due south. Unaware that his speed has been detected, the car driver continues speeding north. The motorcycle policeman slows to a stop at \( t=t_s \), turns around quickly, and gives chase, gradually accelerating to \( 130 \) km/h by \( t = t_1 \), then slowing slightly before overtaking the car at \( t = t_2 \). Both vehicles then pull over to the side of the highway and come to a stop at \( t = t_f \ \), with the motorcycle slightly ahead of the car.
      1. Taking displacement as positive to the north, sketch (on the same axes) qualitative graphs \( y = s_1(t) \) and \( y = s_2(t) \) of the positions of the car and the motorcycle for \( 0 \leq t \leq t_f \), clearly indicating the times \( t = t_1 \) and \( t = t_2 \). Then add to your sketch a graph of the distance \( d \) between the two vehicles.
      2. On a second set of axes directly below the first, sketch qualitative graphs of the velocity, \( v_1(t) \), of the car and of the velocity, \( v_2(t) \), of the motorcycle.
    2. To the right is a graph of a continuous and differentiable function \( y=f(x) \). Polynomial with turning points at x=1, 2.5, 5, negative leading coefficient, zeros at x = 0, 6.2
      1. On a similar set of axes, construct a qualitative graph of the function \[ g(x) = \frac{f(x)-f(1)}{x-1} \] To simplify the construction, place your new axes directly below a copy of the given graph. Explain briefly the key ideas you use.
      2. Add to your diagram in part i) a qualitative sketch of the derivative function \( f'(x) \). How (if at all) are \( g(x) \) and \( f'(x) \) related?
    3. Franz has a bike which has one low gear (easier to pedal, with more pedalling needed to produce one revolution of the wheels) and one high gear (harder to pedal, but less pedalling needed to produce one revolution of the wheels). Thus in low gear, it is easier to get started and gradually increase speed, while in high gear, it is easier to maintain a higher velocity once the bike gets going.

      Two quadratic-like curves; one is initially below the other, but overtakes it after a certain point

      At the right are two qualitative graphs of distance \( x \) versus time \( t \). Each graph represents distance travelled when Franz pedals hard from a standing start so as to reach, as quickly as possible, the top speed in one of the gears.

      1. Decide which graph represents low gear, and which represents high gear. Justify the shape of each graph.
      2. On another set of axes, sketch qualitative graphs of the bike's velocity \( v \) versus time, for each of the two gears.
      3. Suppose Franz wants to attain the maximum possible speed on this bike in the least possible time, making use of both gears. Assuming he can switch gears instantaneously, show how to use your graphs from part ii) to determine at what time \( t^{\text{*}} \) he should switch gears. Then sketch the resulting velocity versus time graph.
  6. Object and image. Enlarged image at distance x to left of convex lens; original image at distance y to right of lens

    For a thin lens, the distance \( x \gt 0 \) between the object and lens, and the distance \( y \gt 0 \) between the image and lens, are related reciprocally to the focal length \( f \) (a positive constant) of the lens, as follows: \[ \frac{1}{x}+ \frac{1}{y} = \frac{1}{f} \]

    1. Show that \( y \gt f \) and \( x \gt f \).
    2. Show that, as the object moves, the image moves in the same direction.
  7. More bounce to the ounce!

    At \( t = 0 \), a curious child drops a Superball (a hard, very bouncy ball) from an apartment balcony \( 20 \) m above the ground. The ball hits the ground, and then bounces repeatedly, losing a bit of speed on each bounce.

    1. Until the first bounce, Newton's law predicts that the ball's height above the ground (measured in metres) will be given by \[ y(t) = -5t^2 + 20 \] for \( 0 \leq t \leq t_1 \), where \( t \) is measured in seconds, and \( t_1 \) is the time of the first bounce. Find the velocity \( v(t_1 ) \) with which the ball hits the ground.
    2. Think about what the ball will do, and sketch a qualitative graph of the ball's height \( y \) versus time \( t \) for \( t \geq 0 \). Then use that graph to help you sketch a qualitative graph of the ball's velocity \( v(t) \).
    3. If the clock is restarted at \( t=0 \) on each bounce, the ball's distance above the ground until the next bounce is predicted by Newton's law to be \[ y(t) = -5t^2 + v_n t \] for \( 0 \leq t \leq t_{n+1} \) and \( n = 1,2,3, \ldots \) where \( t_{n+1} \) is the time lapse until the next bounce. Since the ball loses a bit of its impact speed on each bounce, the initial velocity \( v_n \) for the next bounce will be given by \( v_n = \alpha \left\lvert v(t_n ) \right\rvert \) in each case, for some positive fraction \( \alpha \lt 1 \). For each of \( n = 1,2,3 \), find the value of \( v_n \), \( t_{n+1} \), and the maximum height \( y_n \) attained by the ball between the \( n^{th} \) and \( (n+1)^{st} \) bounce (which occurs when the ball stops, i.e., \( v = \frac{dy}{dt}=0 \)).
    4. Use your results from part c) to predict a sequence for each of \( y_n, n=1,2,3, \ldots \) and \( t_n, n=2,3,4,\ldots \).
    5. Create a geometric series representing the total distance the ball travels, and the total time it takes. Assuming that the constant \( \alpha \) is equal to \( \frac{4}{5} \), find the sum of each series.
  8. Play ball! Parabola opening downward with intercepts (0,0), (R,0); tangent line drawn at x=0, angle theta between tangent and x-axis highlighted

    For an object thrown from the origin \( (0,0) \) at \( t=0 \), with initial speed \( v_0 \gt 0 \), at angle \( \theta \) to the horizontal, Newton's law of motion predicts that its position \( (x(t),y(t)) \) for \( t \gt 0 \) will be given by \begin{align*} x &= (v_0 \cos (\theta)) t \\ y &= (v_0 \sin (\theta)) t - \frac{1}{2} g t^2, \end{align*} where \( g \) is a positive constant called gravitational acceleration.

    1. Find an equation for the path of the object in the form \( y=f(x) \). Note that \( f \) will also depend on the parameters \( v_0\), \(\theta \), and \( g \).
    2. Find the value \( y=h \), the highest point on the path (at which the object stops moving upward before beginning to move downward).
    3. Find the range, \( R \), which is the value of \( x \) when the object returns to \( y=0 \).
    4. When closed, the roof of the Rogers Centre in Toronto rises \( 86 \) m above the centre of the field. Assuming the pitcher's mound is about \( 35 \) m from the centre of the field, and the pitcher releases the ball about \( 2 \) m above the ground, what initial speed \( v_0 \) and angle \( \theta \) would be necessary for a pitch to hit the roof at its highest point? Let \( (0,0) \) be the release point of the ball, and use \( g=9.8 \) m/s2.
    5. Do you think a pitcher could achieve such a pitch?

      NOTE: Most pitchers can throw between \( 40 \) to \( 45 \) m/s horizontally, but much less than that (about \( 30 \) to \( 35 \) m/s) vertically.

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