Question

A box (with a penguin inside) with mass, M, slides down a frictionless ramp, starting a height H above ground level. At
the lowest point of the ramp (height = -0.1H) it slides through a curved section of track with radius, R= 0.25H. The
box (w/ penguin) then rises to ground level (height = 0) and
at that point, leaves a jump at an angle θ. At the highest
point of its trajectory, h2, it strikes a blob of glue (mass=M) hanging on a rope (length, l) – which is at a horizontal distance, d, from the ramp. The box (w/ penguin) sticks to
the blob of glue and together they swing up to a maximum
height, h3. In terms of the variables provided, express the following:

a) The Kinetic Energy and the Centripetal Acceleration of the box (w/ Penguin) at the lowest point of ramp. b) The Height, h2, and the horizontal Distance, d, from the jump to the point of collision.
c) The Speed, v3 after the collision, and the Height, h3
d) The Time to swing up to height, h3 after the collision of blob & box (w/ penguin).

Answer 1

Kinetic energy at end of ramp = (1/2)mv2 = m g (1.1 h )

where m is mass of box, v is speed at point C , g is acceleration due to gravity, h is reference height of starting point from ground level as shown in figure.

Centripetal acceleraion ar at C is given by

where R is radius of curvature of circular track

speed at lowest point D is obtained from

..............................(1)

speed at E is obtained from the following equation

By substituting vD from eqn.(1), we get vE from eqn. (2) as

after jumping from ground level , height h1 reached at F is given by

time t taken to reach height at F is given by

Distance d travelled to reach F is given by

velocity v of combined box after collision

height h2 is obtained from

Time t taken to reach G