PHY 53
Summer 2010
Assignment 2
Reading: Kinematics 2
Key Concepts: Projectiles; circular motion.
[In problems with numbers, use g = 10 m/s2 .]
1.
2.
3.
Ball A and ball B are thrown with the same vertical component of initial velocity,
but ball A is thrown at a 30° angle above the horizontal and ball B at a 60° angle.
They both land at the level from which they were thrown. Neglect air resistance.
Comment on the validity of each statement, and give your reasoning.
a.
Ball B stays in the air longer.
b.
The balls will travel the same distance horizontally before landing.
c.
Ball B will rise higher.
d.
Ball A will land with the greater speed.
A ball on a string is moving with constant speed v in a
circle of radius R on a horizontal surface. As shown from
above, the ball is at the location shown. Comment on the
validity of each statement, and give your reasoning.
a.
The ball’s acceleration is v 2 /R to the right.
b.
If the string breaks at this instant, the ball will
move off to the right.
c.
The tangential acceleration of the ball is zero.
d.
Because the ball’s speed is constant, there is no acceleration.
v
R
Two cannons on a slope are aimed directly at each other
as shown.
a.
b.
Show that if they fire at the same time the shells
will hit each other in flight, if the sum of their
initial speeds is large enough.
θ
Show that if they do not fire at the same time the
shells cannot hit eachother.
[Write the equation for the difference between their displacement vectors.]
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PHY 53
4.
5.
Summer 2010
A girl is standing on a rotating carousel, at distance 4 m from the rotation axis.
She experiences a horizontal acceleration of 0.64 m/s2 .
a.
What is the angular speed of the carousel? Ans: 0.4 rad/s .
b.
What is her linear speed? Ans: 1.6 m/s .
c.
If she moves toward the axis and stands at distance 2 m from it, what is
her acceleration? Ans: 0.32 m/s2 .
d.
What is her new linear speed? Ans: 0.8 m/s .
Two balls are thrown from a vertical cliff of height h, with the same initial speed
but at two different angles, α and β , relative to the horizontal.
a.
Show that they strike the water below with the same speeds.
b.
If α = 35° and β = 50° , which one strikes the water at greater distance
from the base of the cliff? How do you know? [Consult Galileo.]
6.
7*.
A center fielder is trying to throw a runner out at home plate, 70 m from where
he throws the ball. He can throw hard enough to have the ball travel the whole
way in the air, if he throws at a 45° elevation angle.
a.
What initial speed can he give the ball? Ans: 26.5 m/s .
b.
How long does it take the ball to reach the plate? Ans: 3.74 s .
c.
In fact, he chooses to throw the ball so it bounces halfway to the plate. He
throws with the same initial speed, and the ball loses no speed because of
the bounce. At what angle does he throw? [Draw a picture.] Ans: 15° .
d.
How long does it take the ball to reach the plate in this case? Ans: 2.74 s .
A boy whirls a ball attached to a 1 m string in a horizontal circle 2 m above the
ground. The string breaks at the point where the ball is attached; the ball flies off,
landing on the ground 10 m away horizontally. Before the string broke:
a.
What was the ball’s speed? Ans: 15.8 m/s .
b.
What was its radial acceleration? Ans: 250 m/s2 .
c.
What was the period of its motion? Ans: 0.397 s
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PHY 53
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9.
Summer 2010
In WW I the Germans bombarded Paris by means of a very large cannon the
French called “Big Bertha”. Its muzzle velocity was about 1600 m/s.
a.
If it was fired at elevation angle 45° what would be its range? Ans:
256 km .
b.
To what maximum height would the shell rise? Ans: 64 km .
c.
The calculations in (a) and (b) neglect air resistance, which in effect eats
away at the projectile’s speed at every point on the trajectory. Plot the
trajectory in the absence of air resistance, and plot on the same graph a
reasonable guess as to what it would look like with air resistance.
d.
As on would expect, the effect of air resistance is proportional to the
density of air, which decreases rather rapidly with height above the earth.
How might one take advantage of this to increase the range?
An object with a circular crosssection is rolling across a horizontal
floor. You will analyze the motion of
a point on the rim of the object. The
position vector of the point is the
sum of the vector xC to the center of
y
xC
θ
r
the object plus the vector r from
there to the point on the rim, as
shown. The radius of the object is R.
a.
Express the x-coordinate of the point in terms of xC , R and θ .
b.
Take the time derivative to find the horizontal velocity of the point. Use
the notation vC = dxC /dt and ω = dθ /dt .
c.
If the object rolls without slipping then at the instant when the point is in
contact with the floor (i.e., when θ = π / 2 ) its horizontal velocity is zero.
Show that this requires vC = Rω .
x
3