PHY 53
Summer 2010
Assignment 1
Reading: Kinematics 1, Kinematics 2
Key Concepts: Position, velocity, acceleration; trajectory; constant acceleration in 1-D;
vectors, magnitude, direction, components; position, velocity and acceleration in 3-D.
1.
Two cars are traveling
parallel to each other
on a 4-lane highway.
Their positions x as a
function of time t are
shown in the graph,
starting at t = 0 .
a.
2.
x
2
1
Did either of
them start from
rest at t = 0 ? If
so, which
one(s). How do
you know?
t
b.
What is the significance of the point on the graph where the curves cross?
c.
What is the significance of the point on the x-axis where the curve for #1
intersects it?
d.
Did one of them travel at constant speed? Which one? How do you know?
e.
Is there a time at which their speeds were the same? How would you
determine when that was?
The velocity of an object is given by the equation v(t) = 3 − 6t (SI units).
a.
What is the acceleration a(t) ?
b.
If the initial position of the particle is x(0) = 0 , what is x(t) for t ≥ 0 ?
c.
At what time is x = 0 again? Ans: t = 1 s.
d.
Sketch plots of x and v for 0 ≤ t ≤ 2 .
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PHY 53
3*.
Summer 2010
A railroad worker realizes that a car on a slope does not have its brakes set and
will roll down. He runs to the car to set the brakes. At t = 0 , when he is at
distance x0 behind the car, it starts to roll, accelerating at constant rate 2 m/s2 .
He is running at constant speed 4 m/s .
a.
What is the largest value of x0 that will allow him to catch the rolling car?
Ans: x0 = 4 m.
b.
For that value of x0 , at what time t1 does he catch it? Ans: t = 2 s.
c.
If x0 is smaller than the maximum, solving for t1 gives two solutions,
Interpret the larger one. [Sketch the curves of x vs t for both objects.]
4.
A boy at the edge of a cliff of height h = 60 m throws a rock upwards with speed
v0 = 20 m/s; it falls to the beach at the bottom of the cliff. Use g = 10 m/s2 .
a.
How high does it rise above top of the cliff? Ans: 20 m.
b.
What is the total time it is in the air? Ans: 6 s.
c.
With what speed does it land on the beach? Ans: 40 m/s.
d.
Suppose it had been thrown straight down with speed v0 . With what
speed would it land on the beach? Ans: 40 m/s.
5*.
By drawing the arrows and using geometry, prove the following about the sum
and difference of two vectors A and B, whose directions differ by the angle θ .
a.
b.
c.
2
A + B = A 2 + B2 + 2ABcosθ .
The vectors cannot add to zero unless their magnitudes are the same.
2
2
(
)
A + B + A − B = 2 A 2 + B2 .
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PHY 53
6.
7.
Summer 2010
A particle has position vector r(t) = (30t)i + (40t − 5t 2 )j (SI units).
a.
What is its velocity vector? Ans: v(t) = 30i + (40 − 10t)j .
b.
What is its acceleration vector? Ans: a = −10j .
c.
At what time t1 > 0 is it located on the x-axis again? Ans: t1 = 8 s.
d.
Sketch the trajectory from t = 0 to t = t1 .
Two projectiles, 1 and 2, are launched from the same point with the same initial
speed v0 but different elevation angles θ1 and θ 2 . They land at the same level
from which they were launched. Neglect air resistance.
8.
a.
Show that if θ1 > θ 2 , then 1 will stay in flight longer.
b.
Show that if θ1 + θ 2 = π / 2 they will land at the same place.
c.
Show that the maximum horizontal distance traveled occurs if θ = π / 4 .
Mickey gets in his car at a parking lot at Duke and drives to Chapel Hill to pick
up Minnie and bring her to a party at Duke. He parks back at Duke in the same
spot he left, 45 minutes later. His car’s odometer shows he drove a total of 20
miles. After the party he drives Minnie back to Chapel Hill, returning to Duke
three hours later and parking in the same spot. The odometer shows this trip was
also 20 miles. [Give answers in mph.]
a.
What was the average velocity for the first round trip to Chapel Hill?
b.
What was the average velocity for the second round trip to Chapel Hill?
c.
What was the average speed for the first trip?
d.
What was the average speed of the second trip?
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PHY 53
9.
Summer 2010
A monkey is sitting on a tree branch. A hunter on the ground aims his blowgun
directly at the monkey, ready to shoot. The monkey sees the danger and drops
from the branch just as the hunter shoots his dart (at t = 0 ). Take the point at
which the dart emerges from the blowgun as the coordinate origin. Call the
velocity of the dart as it emerges from the blowgun v 0 , and call the initial
displacement of the monkey rm (0) . [These vectors are parallel, because he aimed
directly at the monkey.]
a.
Draw a picture showing the initial situation, and indicate the trajectories.
b.
Write the vector equation describing the location rm (t) of the monkey as
he drops to the ground, in terms of given quantities and g.
c.
Call the displacement of the dart rd . Write the vector equation for rd (t) .
d.
Calculate the displacement of the monkey relative to the dart, rm − rd and
show that (if v0 is large enough) the dart will hit the monkey. [Show that
there is a time when rm − rd = 0 .]
e.
What factors determine how large v0 must be for the dart to hit the
monkey?
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