A block collides into a spring on a frictionless surface... Find v in terms of x. graph v vs. x?

Question

there are no numbers for this problems....
to graph velocity versus displacement is very challenging...

Answer 1

at the moment of impact, momentum will be conserved and the kinetic energy will be converted to energy stored in the spring.

using conservation of energy
.5*k*x^2=.5*m*(v0-v)
where v0 is the speed at impact
so
v=v0-k*x^2/m
since k/m is a constant, let's call it C, and v=v0 when x=0. that is the y intercept
v=v0-C*x^2
this is only valid until v=0, then it will rebound back along the same path until v=v0 at which time the velocity is constant since there is no friction.

Here's a graph where v0=50, C=2, and the impact occurs at x=0 with motion to the right
http://i142.photobucket.com/albums/r88/odu83/spring.jpg


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Answer 2

v^2 = u^2 - 2ax; f = ma - kx> 0; so a > kx/m and v^2 = u^2 - 2(k/m)x^2 as long as there is a net force on the block and the spring continues to compress. ma is the force exerted by the block and kx is the counter force exerted by the spring.

Graph v^2 = u^2 - 2(k/m)x^2; use y = v^2 as the dependent variable and X = x^2 as the independent. k/m is a constant for a given block of mass m and a given spring with a compression coefficient k. b = u^2 is your Y intercept where y = b - 2(k/m)X = -2k/m X + b = -MX + b, which is the general equation for a straight line with M = -(2k/m) as the slope.

Now you can set X = x^2 to a series of independent values, like X = 0, 1, 2, .... , N and plot the resulting y = v^2. But you need to select k and m in order to derive M, the slope. As you plot, keep in mind that one unit of X on the graph is equal to x^2, the square of the spring compression distance. Also one unit of y on that graph is equal to v^2, the square of the velocity.

Answer 3

do your own homework and if you can't talk to your teacher, don't rely on people on the internet who are just guessing to sound smart.