A baseball is hit almost straight up into the air with a speed of . Estimate (a) how high it goes and (b) how long it is in the air. (c) What factors make this an estimate?
(a) The ball goes up to the height of 31.89 m.
(b) The ball stays for 5.1 s in the air.
(c) The acceleration due to gravity and wind resistance can affect the estimation.
The rate of change of velocity that is produced due to the gravitational pull of the earth is called acceleration due to gravity.
The value of acceleration due to gravity is generally taken to be , in the downward direction.
The initial speed of the ball, .
Assume that the maximum height achieved by the ball is h, and the ball stays for time in the air.
The acceleration due to gravity is . (Taking the upward direction as positive)
When the ball reaches its maximum height, its velocity will become zero. Thus, using the third equation of motion,
Substituting the values in the above equation,
Solving for the value of h,
Thus, the maximum height achieved by the ball is 31.89 m.
From the second equation of motion, you can write the displacement of the ball at time t as
Here, as the initial and final positions are the same, the displacement of the ball is zero. Thus,
Solving the above quadratic equation and neglecting root t = 0,
Thus, the total time for which the ball was in the air is 5.1 s.
In a putting game, the force with which a golfer strikes a ball is planned so that the ball will stop within some small distance of the cup, say 1.0 m long or short, in case the putt is missed. Accomplishing this from an uphill lie (that is, putting the ball downhill, see Fig. 2–47) is more difficult than from a downhill lie. Assume that on a particular green, the ball constantly decelerates at going downhill and at going uphill to see why. Suppose you have an uphill lie 7.0 m from the cup. Calculate the allowable range of initial velocities you may impart to the ball so that it stops in the range 1.0 m short to 1.0 m long of the cup. Do the same for a downhill lie 7.0 m from the cup. What, in your results, suggests that the downhill putt is more difficult?
FIGURE 2-47 Problem 70
Pelicans tuck their wings and free-fall straight down while diving for fish. Suppose a pelican starts its dive from a height of 14.0 m and cannot change its path once committed. If it takes a fish 0.20 s to perform an evasive action, at what minimum height must it spot the pelican to escape? Assume that the fish is at the surface of the water.
Digital bits on an audio CD of diameter 12.0 cm are encoded along an outward spiraling path that starts at radius and finishes at radius . The distance between the centers of the neighbouring spiral windings is .
(a) Determine the total length of the spiral into a straight path [Hint: Imagine ‘unwinding’ the spiral and the straight path of width , and note that the original spiral and the straight path both occupy the same area.]
(b) To read information, a CD player adjusts the rotation of the CD so that the player’s readout laser moves along the spiral path at a constant speed of about 1.2 m/s. Estimate the maximum playing time of such a CD.
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