Jane and Michael are inventing a game using a spring and marble of mass 20 0.0 g

Question

Jane and Michael are inventing a game using a spring and marble of mass 200.0 g in which they plan on employing inclined planes. As they carefully plan out the design, they decide to launch the marble along an incline of theta=15o using the spring of spring constant k=2.10 N/cm as shown in the figure.
The plan is to compress the spring by 7.00 cm and launch the marble along the incline. In order to complete the design they need to know the speed of the marble just as it leaves the spring and the final distance the marble will travel up the incline.
a) Find an expression for the speed of the marble just as it leaves the spring. Hint: Do not forget about the work done by the gravitational force and assume that the marble slides without friction b) Calculate a numerical value for the speed of the marble just as it leaves the spring. c) Find the distance up the incline that the marble will travel before coming to rest momentarily. Jane and Michael are inventing a game using a spring and marble of mass 200.0 g in which they plan on employing inclined planes. As they carefully plan out the design, they decide to launch the marble along an incline of theta=15o using the spring of spring constant k=2.10 N/cm as shown in the figure.
The plan is to compress the spring by 7.00 cm and launch the marble along the incline. In order to complete the design they need to know the speed of the marble just as it leaves the spring and the final distance the marble will travel up the incline.
a) Find an expression for the speed of the marble just as it leaves the spring. Hint: Do not forget about the work done by the gravitational force and assume that the marble slides without friction b) Calculate a numerical value for the speed of the marble just as it leaves the spring. c) Find the distance up the incline that the marble will travel before coming to rest momentarily.

Explanation / Answer

a)Conserving energy,

potential energy of spring=gravitational potential energy+kinetic energy of ball

0.5kx^2=mgdSin (theta)+0.5mv^2

Just as the marble leaves the spring, the compression in the spring becomes zero. Note that d=0.07m=length of spring compression.

v=sqrt(0.5kx^2-2gdSin (theta)).

b)v=0.40m/s

c)Conserving energy,

0.5mv^2=mgdSin 15

d=v^2/(2gSin (15))=3.14cm

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