A dinosaur fossil is slowly moving down the slope of a glacier
under the influence of wind, rain and gravity. At the same time,
the glacier is moving relative to the continent underneath. The
dashed lines represent the directions but not the magnitudes of the
velocities. Pick a scale, and use graphical addition of vectors to find
the magnitude and the direction of the fossil's velocity relative to
the continent. You will need a ruler and protractor.
Is it possible for a helicopter to have an acceleration due east
and a velocity due west? If so, what would be going on? If not, why
A bird is initially flying horizontally east at 21.1 m/s, but one
second later it has changed direction so that it is flying horizontally
and 7 ° north of east, at the same speed. What are the magnitude
and direction of its acceleration vector during that one second time
interval? (Assume its acceleration was roughly constant.)
A person of mass M stands in the middle of a tightrope, which is
fixed at the ends to two buildings separated by a horizontal distance
L. The rope sags in the middle, stretching and lengthening the rope
(a) If the tightrope walker wants the rope to sag vertically by no
more than a height h, find the minimum tension, T, that the rope
must be able to withstand without breaking, in terms of h, g, M,
(b) Based on your equation, explain why it is not possible to get
h = 0, and give a physical interpretation.
Your hand presses a block of mass m against a wall with a
force FH acting at an angle θ. Find the minimum and maximum
possible values of |FH| that can keep the block stationary, in terms
of m, g, θ, and µs, the coefficient of static friction between the block
and the wall.
A skier of mass m is coasting down a slope inclined at an angle
θ compared to horizontal. Assume for simplicity that the treatment
of kinetic friction given in chapter 5 is appropriate here, although a
soft and wet surface actually behaves a little differently. The coefficient
of kinetic friction acting between the skis and the snow is µk,
and in addition the skier experiences an air friction force of magnitude
bv2, where b is a constant.
(a) Find the maximum speed that the skier will attain, in terms of
the variables m, g, θ, µk, and b.
(b) For angles below a certain minimum angle θmin, the equation
gives a result that is not mathematically meaningful. Find an equation
for θmin, and give a physical explanation of what is happening
for θ < θmin.
A gun is aimed horizontally to the west, and fired at t = 0. The
bullet's position vector as a function of time is
where b, c, and d are constants.
(a) What units would b, c, and d need to have for the equation to
(b) Find the bullet's velocity and acceleration as functions of time.
(c) Give physical interpretations of b, c, d, ´
Annie Oakley, riding north on horseback at 30 mi/hr, shoots
her rifle, aiming horizontally and to the northeast. The muzzle speed
of the rifle is 140 mi/hr. When the bullet hits a defenseless fuzzy
animal, what is its speed of impact? Neglect air resistance, and
ignore the vertical motion of the bullet.
Solution, p. 281
A cargo plane has taken off from a tiny airstrip in the Andes,
and is climbing at constant speed, at an angle of θ=17 °with respect
to horizontal. Its engines supply a thrust of Fthrust = 200 kN, and
the lift from its wings is Flift = 654 kN. Assume that air resistance
(drag) is negligible, so the only forces acting are thrust, lift, and
weight. What is its mass, in kg?
Solution, p. 281
A wagon is being pulled at constant speed up a slope θ by a
rope that makes an angle φ with the vertical.
(a) Assuming negligible friction, show that the tension in the rope
is given by the equation
where FW is the weight force acting on the wagon.
(b) Interpret this equation in the special cases of φ = 0 and φ =
180 ° - θ.
Solution, p. 282
The angle of repose is the maximum slope on which an object
will not slide. On airless, geologically inert bodies like the moon or
an asteroid, the only thing that determines whether dust or rubble
will stay on a slope is whether the slope is less steep than the angle
(a) Find an equation for the angle of repose, deciding for yourself
what are the relevant variables.
(b) On an asteroid, where g can be thousands of times lower than
on Earth, would rubble be able to lie at a steeper angle of repose?
Solution, p. 282
The figure shows an experiment in which a cart is released
from rest at A, and accelerates down the slope through a distance
x until it passes through a sensor's light beam. The point of the
experiment is to determine the cart's acceleration. At B, a cardboard
vane mounted on the cart enters the light beam, blocking the
light beam, and starts an electronic timer running. At C, the vane
emerges from the beam, and the timer stops.
(a) Find the final velocity of the cart in terms of the width w of
the vane and the time tb for which the sensor's light beam was
(b) Find the magnitude of the cart's acceleration in terms of the
measurable quantities x, tb, and w.
(c) Analyze the forces in which the cart participates, using a table in
the format introduced in section 5.3. Assume friction is negligible.
(d) Find a theoretical value for the acceleration of the cart, which
could be compared with the experimentally observed value extracted
in part b. Express the theoretical value in terms of the angle θ of
the slope, and the strength g of the gravitational field.
Problem 13 (Millikan and Gale,
The figure shows a boy hanging in three positions: (1) with
his arms straight up, (2) with his arms at 45 degrees, and (3) with
his arms at 60 degrees with respect to the vertical. Compare the
tension in his arms in the three cases.