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Question

The potential energy (in joules) of a particle of mass 1 kg moving in a plane is given by U=3x+4y. The position coordinates of the point being x and y, measured in metres. If the particle is at rest at (6,4); then

A
its acceleration is of magnitude 5 m/s2
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B
its speed when it crosses the yaxis is 10 m/s
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C
it crosses the y axis (x=0) at y=4
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D
it moves in a straight line passing through the origin (0,0)
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Solution

The correct option is C it crosses the y axis (x=0) at y=4
Given: U=3x+4y
We know that,
F=(dUdx^i+dUdy^j+dUdz^k)
F=(3^i+4^j) N
F=(3^i4^j) N
Acceleration of particle=Fm=(3^i4^j) m/s2
a=5 m/s2
Let at time t the particle cross y axis
Sx=12(ax)t26=12(3)t2t=2 sec
Along y direction:
Sy=12(4)(2)2=8
y4=8
Particle crosses y-axis at y=4
At (6,4):U=34 J & KE=0
At (0,4):U=16 JKE16=34 J
12mv2=50v=10 m/s

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