There are (one can say) three coequal theories of motion for a single particle:

Question

There are (one can say) three coequal theories of motion for a single particle: Newton's second law, stating that the total force on an object causes its acceleration; the work–kinetic energy theorem, stating that the total work on an object causes its change in kinetic energy; and the impulse–momentum theorem, stating that the total impulse on an object causes its change in momentum. In this problem, you compare predictions of the three theories in one particular case. A 2.10-kg object has velocity 7.00 m/s. Then, a constant net force 9.0î N acts on the object for 4.50 s.

(e) Find the work done on the object from

(g) Find the final kinetic energy from .5mvi2 + W

Explanation / Answer

a) Impulse = change in momentum

F*t = m*(vf - vi)

F*t/m = vf - vi

==> vf = vi + F*t/m

= 7 j + 9*4.5/2.1 i

= ( 19.28 i + 7 j ) m/s

b) a = (vf - vi)/t

= (19.28 i + 7j - 7j)/4.5

= 4.28 i m/s^2

c) a = F/m

= 9/2.1

= 4.28 i m/s^2

d) delta_r = vi*t + (1/2)*a*t^2

= 7*4.5 j + (1/2)*4.28*4.5^2 i

= (43.3 i + 31.5 j ) m

e) W = F.delta_r

= (9i).(43.3 i + 31.5 j)

= 9*43.3

= 390 J

f) final KE = (1/2)*m*vf.vf

= (1/2)*2.1*(19.28 i + 7j).((19.28 i + 7j)

= (1/2)*2.1*(19.28^2 + 7^2)

= 441.8 J

g) final KE = (1/2)*m*vi^2 + W

= (1/2)*2.1*7^2 + 390

= 441.8 J

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