Hooke's law describes an ideal spring. Many real springs are better described by the restoring force (F Sp ) s =−kΔs−q(Δs) 3 (p)s=−kΔs−q(Δs)3 , where q q is a constant. Consider a spring with k kk = 350 N/m N/m and q qq = 750 N/m 3 N/m3 .How much work must you do to compress this spring 15cm? Note that, by Newton's third law, the work you do on the spring is the negative of the work done by the spring. By what percent has the cubic term increased the work over what would be needed to compress an ideal spring?

Answer with Explanation:

We are given that

Restoring force,

We have to find the work must you do to compress this spring 15 cm.

Using 1 m=100 cm

Work done=

W=

Ideal spring work=

Percentage increase in work=%

the velocity you are looking for is made up of two parts: horizontally velocity, vh, and its vertical velocity, vv. the actual velocity of the ball will be these two parts added together, and because velocity is a vector then they will need to be added vectorially.  

neglecting air resistance etc etc.,  

the horizontal velocity component will be constant (because there is no accelerating force in the horizontal direction.  

vh = 8 ms-1  

the vertical velocity component has an accelerating force: 'g'.  

using v = u +at where a = g = 9.81 ms-2.  

the vertical component of velocity, vv = 0 + gt = vv =9.81 x 2  

vv = 19.62 ms-1 . the zero appears because the virtical component of velocity at time t=0 is zero. remember, the ball was thrown horizontally.  

how fast it will be moving (it actual path (being a parabola )) will be the vector sum of these two.  

adding vectorally gives: -  

v = sqr root (8^2 + 19.62^2) = sqr root(64+384.9) = 21.18 or 21.2 ms-1