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1.1. HOW DO WE MEASURE VELOCITY?
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10
15
20
50
100
150
200
s
t
Figure 1.3: A bungee jumper’s height function.
(e) Among the average and instantaneous velocities you computed in earlier
questions, which are positive and which are negative? What does negative
velocity indicate?
2. A diver leaps from a 3 meter springboard. His feet leave the board at time t = 0,
he reaches his maximum height of 4.5 m at t = 1.1 seconds, and enters the water at
t = 2.45. Once in the water, the diver coasts to the bottom of the pool (depth 3.5 m),
touches bottom at t = 7, rests for one second, and then pushes off the bottom. From
there he coasts to the surface, and takes his first breath at t = 13.
(a) Let s(t) denote the function that gives the height of the diver’s feet (in meters)
above the water at time t. (Note that the “height” of the bottom of the pool
is −3.5 meters.) Sketch a carefully labeled graph of s(t) on the provided axes
in Figure 1.4. Include scale and units on the vertical axis. Be as detailed as
possible.
(b) Based on your graph in (a), what is the average velocity of the diver between
t = 2.45 and t = 7? Is his average velocity the same on every time interval
within [2.45, 7]?
(c) Let the function v(t) represent the instantaneous vertical velocity of the diver
at time t (i.e. the speed at which the height function s(t) is changing; note
that velocity in the upward direction is positive, while the velocity of a falling
object is negative). Based on your understanding of the diver’s behavior, as
well as your graph of the position function, sketch a carefully labeled graph of
v(t) on the axes provided in Figure 1.4. Include scale and units on the vertical
axis. Write several sentences that explain how you constructed your graph,
discussing when you expect v(t) to be zero, positive, negative, relatively large,
and relatively small.
1.1. HOW DO WE MEASURE VELOCITY?
5
10
15
20
50
100
150
200
s
t
Figure 1.3: A bungee jumper’s height function.
(e) Among the average and instantaneous velocities you computed in earlier
questions, which are positive and which are negative? What does negative
velocity indicate?
2. A diver leaps from a 3 meter springboard. His feet leave the board at time t = 0,
he reaches his maximum height of 4.5 m at t = 1.1 seconds, and enters the water at
t = 2.45. Once in the water, the diver coasts to the bottom of the pool (depth 3.5 m),
touches bottom at t = 7, rests for one second, and then pushes off the bottom. From
there he coasts to the surface, and takes his first breath at t = 13.
(a) Let s(t) denote the function that gives the height of the diver’s feet (in meters)
above the water at time t. (Note that the “height” of the bottom of the pool
is −3.5 meters.) Sketch a carefully labeled graph of s(t) on the provided axes
in Figure 1.4. Include scale and units on the vertical axis. Be as detailed as
possible.
(b) Based on your graph in (a), what is the average velocity of the diver between
t = 2.45 and t = 7? Is his average velocity the same on every time interval
within [2.45, 7]?
(c) Let the function v(t) represent the instantaneous vertical velocity of the diver
at time t (i.e. the speed at which the height function s(t) is changing; note
that velocity in the upward direction is positive, while the velocity of a falling
object is negative). Based on your understanding of the diver’s behavior, as
well as your graph of the position function, sketch a carefully labeled graph of
v(t) on the axes provided in Figure 1.4. Include scale and units on the vertical
axis. Write several sentences that explain how you constructed your graph,
discussing when you expect v(t) to be zero, positive, negative, relatively large,
and relatively small.