EXAMPLE 5 Motion in a gravitational field A stone is launched vertically upward with a velocity of v m/s from a point s, meters above the ground, where vo > 0 and S, 2 0. Assume the stone is launched at time t = 0 and that s(t) is the position of the stone at time t > 0; the positive s-axis points upward with the origin at the ground. By Newton's Second Law of Motion, assuming no air resistance, the position of the stone is governed by the differential equation s"(t) = -8, where g = 9.8 m/s² is the accelera- tion due to gravity (in the downward direction). a. Find the position s(t) of the stone for all times at which the stone is above the ground. b. At what time does the stone reach its highest point and what is its height above the ground? c. Does the stone go higher if it is launched at v(0) = vo = 39.2 m/s from the ground (So = 0) or at vo = 19.6 m/s from a height of so = 50 m?

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Suppose the initial conditions in the given example part (a) are ν0 = 14.7 m/s and s0 = 49 m. Write the position function s(t), and state its domain. At what time will the stone reach its maximum height? What is the maximum height at that time?

EXAMPLE 5 Motion in a gravitational field A stone is launched vertically upward
with a velocity of v m/s from a point s, meters above the ground, where vo > 0 and
S, 2 0. Assume the stone is launched at time t = 0 and that s(t) is the position of the
stone at time t > 0; the positive s-axis points upward with the origin at the ground. By
Newton's Second Law of Motion, assuming no air resistance, the position of the stone is
governed by the differential equation s"(t) = -8, where g = 9.8 m/s² is the accelera-
tion due to gravity (in the downward direction).
a. Find the position s(t) of the stone for all times at which the stone is above the ground.
b. At what time does the stone reach its highest point and what is its height above the
ground?
c. Does the stone go higher if it is launched at v(0) = vo = 39.2 m/s from the ground
(So = 0) or at vo = 19.6 m/s from a height of so = 50 m?
Transcribed Image Text:EXAMPLE 5 Motion in a gravitational field A stone is launched vertically upward with a velocity of v m/s from a point s, meters above the ground, where vo > 0 and S, 2 0. Assume the stone is launched at time t = 0 and that s(t) is the position of the stone at time t > 0; the positive s-axis points upward with the origin at the ground. By Newton's Second Law of Motion, assuming no air resistance, the position of the stone is governed by the differential equation s"(t) = -8, where g = 9.8 m/s² is the accelera- tion due to gravity (in the downward direction). a. Find the position s(t) of the stone for all times at which the stone is above the ground. b. At what time does the stone reach its highest point and what is its height above the ground? c. Does the stone go higher if it is launched at v(0) = vo = 39.2 m/s from the ground (So = 0) or at vo = 19.6 m/s from a height of so = 50 m?
Expert Solution
Step 1

Suppose the initial conditions in the given example part (a) are ν0 = 14.7 m/s and s0 = 49 m.

To write the position function s(t), and state its domain. At what time t will the stone reach its maximum height h? What is the maximum height at that time

Acceleration due to gravity, g=9.8 ms2

Taking the downward direction as negative y-axis.

(a) s''t=-g

Integrating once,

s'(t)=-gt+c1

Which is velocity,

v(t)=s'(t)=-gt+c1    [1]

Integrating again,

s(t)=-gt22+c1t+c2    [2]

Using initial conditions to find the value of the constants c1 and c2.

At t = 0, s0 = 49 m

s(0)=-g022+c1.0+c2=49 mc2=49 m

at t = 0, v0 =14.7 m/s

v(0)=-g.0+c1=14.7 m/sc1=14.7 m/s

So, [2]  becomes,

s(t)=-gt22+14.7t+49m   [3]

(b) t=?

Velocity at maximum height h is v = 0 m/s

v-v0=-gt0-14.7 ms=-9.8 ms2tt=1.5 s

At t = 1.5 s, s(t) = h, from [3],

s(1.5 s)=-9.8 ms21.5 s22+14.71.5 s+49mh=60 m

(c) For v(0) = 39.2 m/s, x(0) = 0 m

v-v0=-gt0-39.2 ms=-9.8 ms2tt=4 s

Using initial condition in [1],

v(0)=-g.0+c1=39.2 m/sc1=39.2 m/s

Using initial condition in [2],

s(0)=-g022+c1.0+c2=0c2=0 m

Equation [2]  becomes,

s(t)=-gt22+39.2tm

At t = 4 s

s(4 s)=-9.8 ms24 s22+39.2 ms4 sm=78.4 m

For v(0) = 19.6 m/s, x(0) = 50 m

v-v0=-gt0-19.6 ms=-9.8 ms2tt=2 s

Using initial condition in [1],

v(0)=-g.0+c1=19.6 m/sc1=19.6 m/s

Using initial condition in [2],

s(0)=-g022+c1.0+c2=50 mc2=50 m

Equation [2]  becomes,

s(t)=-gt22+19.6t+50m

At t = 2 s

s(2 s)=-9.8 ms22 s22+19.6 ms2 s+50m=69.6 m

Yes, the stone goes higher in both the cases.

 

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