Vector Mechanics for Engineers: Statics and Dynamics
Vector Mechanics for Engineers: Statics and Dynamics
12th Edition
ISBN: 9781259638091
Author: Ferdinand P. Beer, E. Russell Johnston Jr., David Mazurek, Phillip J. Cornwell, Brian Self
Publisher: McGraw-Hill Education
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Chapter 13.3, Problem 13.150P

A 180-lb man and a 120-lb woman stand at opposite ends of a 300-lb boat, ready to dive, each with a 16-ft/s velocity relative to the boat. Determine the velocity of the boat after they have both dived, if (a) the woman dives first, (b) the man dives first.

Fig. P13.150

Chapter 13.3, Problem 13.150P, A 180-lb man and a 120-lb woman stand at opposite ends of a 300-lb boat, ready to dive, each with a

(a)

Expert Solution
Check Mark
To determine

Find the velocity of the boat (v2) if woman dives first.

Answer to Problem 13.150P

The velocity of the boat (v2) if woman dives first is 2.80ft/s_.

Explanation of Solution

Given information:

The weight of the man (Wm) is 180lb.

The weight of the woman (Ww) is 120lb.

The weight of the boat (Wb) is 300lb.

The relative velocity of the boat (vr) is 16ft/s.

The acceleration due to gravity (g) is 32.2ft/s2.

Calculation:

Calculate the mass of the man (mm) using the formula:

mm=Wmg

Substitute 180lb for Wm.

mm=180lbg

Calculate the mass of the woman (mw) using the formula:

mw=Wwg

Substitute 120lb for WB.

mw=120lbsg

Calculate the velocity of the boat when woman dives first (v2) using the relation:

Show the free body momentum diagram of the boat system after the woman dives as in Figure (1).

Vector Mechanics for Engineers: Statics and Dynamics, Chapter 13.3, Problem 13.150P , additional homework tip  1

Calculate the velocity of the diving of the woman (vw) with respect to relative velocity of boat using the relation:

vw=vrv1

Here, v1 is the velocity of the boat after the woman dives.

Substitute 16ft/s for vr.

vw=16ft/sv1

Calculate the initial momentum [(P1)w] of the woman before diving using the relation:

(P1)w=mwvw

Substitute 16v1 for vw and 120lbsg for mw.

(P1)w=(120lbsg)(16v1)

The final velocity of the woman after diving is zero, so the final momentum of the woman (P2)w is zero.

Calculate the momentum of boat and man after the woman dives using the relation:

(P)m+b=mm+bv1

Here, (P)m+b is the momentum of both man and boat and mm+b is the mass of both the man and boat.

Substitute 300+180lbsg for mm+b.

(P)m+b=(300+180lbsg)v1

The expression for principle of conservation of momentum as follows;

(P1)w+(P)m+b=(P2)w

Substitute (120lbsg)(16v1) for (P1)w, (300+180lbsg)v1 for (P)m+b, and 0 for (P2)w.

(120lbsg)(16v1)(300+180lbsg)v1=0(120g)(16v1)=(300+180g)v1(16v1)v1=480120

(16v1)v1=416v1=4v15v1=16v1=3.20ft/s(towards the right)

Calculate the velocity of the boat after the man dives (v2) using the relation:

Show the free body momentum diagram of the boat after man dives as in Figure (2).

Vector Mechanics for Engineers: Statics and Dynamics, Chapter 13.3, Problem 13.150P , additional homework tip  2

Calculate the velocity of the diving of the man (vm) with respect to relative velocity of boat using the relation:

vm=vrv2

Substitute 16ft/s for vr.

vm=16ft/sv2

Calculate the initial momentum [(P1)m] of the man before diving using the relation:

(P1)m=mmvm

Substitute 16v2 for vw and 180lbsg for mw.

(P1)m=(180lbsg)(16v2)

Calculate the momentum of the boat after the woman has dived using the relation:

(P)m+b=(300+180lbsg)v1

Calculate the final momentum of the boat after the man has dived [(P)b] using the relation:

(P)b=mbv2

Substitute 300lbsg for mb.

(P)b=(300g)v2

The expression for the principle of conservation of momentum to the man and boat as follows;

(P)m+b=(P2)b+(P1)m

Substitute (300+180lbsg)v1 for (P)m+b, (300g)v2 for (P2)b, and (180lbsg)(16v2) for (P1)m.

(300+180lbsg)v1=(300g)v2+(180lbsg)(16v2)480v1=2,880180v2300v2480v2=2,880480v1

Substitute 3.20ft/s for v1.

480v2=2,880480(3.20)480v2=2,8801,536v2=1,344480v2=2.80ft/s

Therefore, the velocity of the boat (v2) if woman dives first is 2.80ft/s_.

(b)

Expert Solution
Check Mark
To determine

Find the velocity of the boat (v2') if man dives first.

Answer to Problem 13.150P

The velocity of the boat (v2') if man dives first is 0.229ft/s_.

Explanation of Solution

Given information:

The weight of the man (Wm) is 180lb.

The weight of the woman (Ww) is 120lb.

The weight of the boat (Wb) is 300lb.

The relative velocity of the boat (vr) is 16ft/s.

The acceleration due to gravity (g) is 32.2ft/s2.

Calculation:

Show the free body momentum diagram of the boat system after the man dives as in Figure (3).

Vector Mechanics for Engineers: Statics and Dynamics, Chapter 13.3, Problem 13.150P , additional homework tip  3

Calculate the velocity of the diving of the man (vm) with respect to relative velocity of boat using the relation:

vm=vrv1

Here, v1 is the velocity of the boat after the man dives.

Substitute 16ft/s for vr.

vm=16ft/sv1

Calculate the initial momentum [(P1)m] of the man before diving using the relation:

(P1)m=mmvm

Substitute 16v1 for vm and 180lbsg for mm.

(P1)m=(180lbsg)(16v1)

The final velocity of the man after diving is zero, so the final momentum (P2)m of the man is zero.

Calculate the momentum of boat and woman after the man dives using the relation:

(P)w+b=mw+bv1

Substitute 300+120lbsg for mw+b.

(P)w+b=(300+120lbsg)v1

The expression for principle of conservation of momentum to the man as follows;

(P1)m+(P)w+b=(P2)m

Substitute (180lbsg)(16v1) for (P1)m, (300+120lbsg)v1 for (P)w+b, and 0 for (P2)m.

(180lbsg)(16v1)(300+120lbsg)v1=0180(16v1)=420v12,880180v1=420v1600v1=2,880v1=4.8ft/s(towards left)

Show the free body momentum diagram of the boat after man dives as in Figure (4).

Vector Mechanics for Engineers: Statics and Dynamics, Chapter 13.3, Problem 13.150P , additional homework tip  4

Calculate the velocity of the diving of the woman (vw) with respect to relative velocity of boat using the relation:

vw=vrv2

Here, v2 is the velocity of the boat after the woman dives.

Substitute 16ft/s for vr.

vw=16ft/sv2

Calculate the initial momentum [(P1)w] of the woman before diving using the relation:

(P1)w=mwvw

Substitute 16v2 for vw and 120lbsg for mw.

(P1)w=(120lbsg)(16v2)

Calculate the final momentum of the boat after the woman has dived [(P)b] using the relation:

(P)b=mbv2

Substitute 300lbsg for mb.

(P)b=(300g)v2

Calculate the velocity of the boat after the woman dives (v2):

(P)w+b=(P2)b+(P1)w

Substitute (300+120lbsg)v1 for (P)w+b, (300g)v2 for (P)b, and (120lbsg)(16v2) for (P1)w.

(300+120lbsg)v1=(300g)v2(120lbsg)(16v2)420v1=300v21,920+120v2420v1=420v21,920

Substitute 4.80ft/s for v1.

420(4.80ft/s)=420v21,920420v21,920=2016420v2=96v2=96420v2=0.229ft/s

Therefore, the velocity of the boat (v2') if man dives first is 0.229ft/s_.

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Chapter 13 Solutions

Vector Mechanics for Engineers: Statics and Dynamics

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