Some kids have built a pellet launcher consisting of an ideal, massless spring (

Question

Some kids have built a pellet launcher consisting of an ideal, massless spring (k) placed inside a frictionless tube attached to a ramp (see diagram). Both tube and ramp are inclined at an angle of q above the horizontal, but unlike the tube, the ramp

Some kids have built a pellet launcher consisting of an ideal, massless spring (k) placed inside a frictionless tube attached to a ramp (see diagram). Both tube and ramp are inclined at an angle of q above the horizontal, but unlike the tube, the ramp's surface does have friction (mk). The tube's bottom end is plugged. The relaxed spring length fits exactly into the remaining space inside the tube. The kids compress the spring by a distance x, place the pellet in the tube and release the spring. The pellet is ejected out of the tube, then it slides (without rolling) up the ramp an additional distance of d, and then it becomes a projectile. The pellet mass = m1. The target is a ball (m2), hanging vertically (at rest) from a mass- less thread (length L), on a nearby tree. At the peak of its motion as a projectile, the pellet hits the ball and bounces off at a speed of v, in the direction opposite to its impact velocity. The ball swings upward in a circular arc. What angle does the thread make (measured from the vertical) at the highest point in the ball's swing?

Explanation / Answer

we know that

(1/2)mv^2 = (1/2)kd^2

or d^2 = (m/k) x v^2

or d = (m/k)^1/2 x v

where d is distance the spring is compressed,m is mass of ball,k is spring constant of spring and v is speed of ball

Also,

d = L x theta

or theta = (d/L) radians = (d/L) x (180/pi) degrees

where L is length of thread

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