CT87 A Block Initially At Rest Is Allowed To

ct8-7. a block initially at rest is allowed to slide down a frictionless ramp and attains a speed v at the bottom. to achieve a speed 2v at t

CT8-7. A block initially at rest is allowed to slide down a
frictionless ramp and attains a speed v at the bottom. To achieve a
speed 2v at the bottom, how many times as high must the new ramp be?
Pink: Green: 2 Blue: 3
Yellow: 4 Purple: none of these.
Answer: 4 times as high. By conservation of energy, PEi = KEf, mgh =
(1/2) mv2.
So, h is proportional to v2. If v doubles, v2 quadruples and so does
h.
CT8-8. A mass slides down a frictionless ramp of height h and hits a
carpet with kinetic friction coefficient K = 1.0. Its initial speed
is zero. How far does the mass slide along the carpet?
P
ink: h Yellow: Less than h Purple: more than h
Answer: h. Use conservation of energy: Initial energy at top of hill =
mgh = energy at bottom of hill (before puck hits rug) = (1/2)m2 = heat
generated by friction as puck slows and stops = magnitude of work done
by friction = . So we have the equation mgh = Kmgx or h = Kx
= x (since K=1).
CT8-9. An object moves along the x-axis . The potential energy U(x)
vs. position x is shown below.

When the object is at postion x=A, which of the following statements
must be true.
Pink: The velocity is negative, i.e. along the -x direction.
Yellow: The acceleration is negative.
Green: The total energy is negative.
Blue: The kinetic energy is positive.
Purple: None of these statements is always true.
Answer: The acceleration is negative. In 1D, F = - dU/dx. The slope of
the curve dU/dx is positive at x=A, so F = ma = -dU/dx is negative
there. The force F and the acceleration a = F/m are negative at x=A
(meaning their directions are to the left, toward negative x.)
The kinetic energy might be zero at x=A, since the total energy of the
system was never specified. That is, the total energy might be such
that x=A is a turning point.