Group A problem
- A block slides without friction at a speed
of v_1 = 15 m/s towards a frictionless ramp.
The block slides up the ramp and then shoots into the
air, eventually landing on a plateau which is level
with the top of the ramp. The ramp has a length,
L, of 12.5 meters and has an angle of
inclination of theta = 30 degrees with respect to
the floor. Your job is to find the formula
for the distance along the plateau where the block
lands (measured relative to the end of the ramp of
course) in terms of v_i, L, g,
and theta. Then plug in the numbers to see what you get.
- Use your formula to make a plot which shows the
distance for the block's landing as a function of
the length of the ramp. What happens for the case
where L is about 23 meters long?
Group B problem
An action movie requires a stunt in which the
great actor "Arnie" fires a projectile from a
flare gun. At the time, Arnie is standing on
top of a truck moving on a horizontal road at a
speed of v_t = 45 miles/hour. He fires
the gun in the direction of nefarious evildoers
who are chasing the truck. The projectile leaves
the gun with a velocity of 250 m/s (relative to
the gun) and the barrel of the flare gun is at
an angle of 30 degrees to the horizontal and
3 meters above the ground. Ignore air resistance.
- If you mark the horizontal position of the
flare gun just at the instant the projectile is
fired and call that your origin, how far
from that point does the projectile land?
(Be careful about relative quantities here!).
- Make a plot of the range of the projectile as
a function of the angle of firing. At what angle
should Arnie fire the flare gun to get the maximum
range? How can you prove it?
Group D problem
A cannon sits at the bottom of a steep hill. The hill
makes a 30 degree angle to the horizontal. The person
firing the cannon can adjust the angle, theta,
which the cannon makes with the horizontal, but the
speed of cannonballs as they leave the cannon is fixed
at v_0. Assuming no air resistance, make a
plot of the distance up the hill at which the
cannonball lands as a function of theta. At what angle
theta should the cannon be set so as to make the ball land
as far up the hill as possible? How do you know you have
the maximum range up the hill for your value of theta?
Group E problem
A child named Kim is sitting in a train, playing with a toy ball.
Kim is so absorbed in her game that she doesn't realize the
train is moving forward with speed 20 m/s. Kim thinks the train
is still stopped in the station. According to Kim, she's
throwing her ball straight up, and then catching it at the same
place she threw it. Kim notices that exactly 1.5 seconds elapse
between her throw and her catch.
The train's windows are big and surprisingly clean. Standing on
the ground at the side of the railroad tracks is Joe. As the
train speeds past, Joe watches Kim throw and catch her ball.
- Suppose Kim is a prodigy in physics.
According to Kim, how fast does she throw her
ball upward?
- According to Kim, how high into the air did the ball go
above her hand, when the ball reached its peak? According
to Joe, how high into the air did the ball go above
Kim's hand, when the ball reached its peak?
- According to Joe, what was the velocity of Kim's ball
immediately after she threw it (remember to specify
velocity you need magnitude AND direction).
- According to Joe, Kim's ball does not travel straight
up and down. Plot the path of Kim's ball as seen by
Joe. According to Joe, what is the distance d between
where Kim throws her ball and where she catches it
according to Joe.
Group F problem
A medieval army is attacking a castle with very tall walls,
100 meters high. The army's cannon is entrenched exactly 50
meters from the castle. The Head Knight decides that the cannon
can cause the most damage to the castle if cannon balls are
fired over the castle wall. Specifically, the head knight wants
the cannon ball's trajectory to be such that its "peak" (i.e. the
cannon ball's highest point) is reached when the cannon ball
is directly over the wall.
The cannon fires balls at 80 m/s. For calculational simplicity,
let's say the acceleration due to gravity is 10 m/s and
ignore frictional effects.
- At what angle theta to the ground should the Head
Knight align the cannon? What is the distance
beyond the wall at which the cannonball lands?
- If the Head Knight failed to read Galileo's treatise on
motion and aligns the cannon at an angle theta of
60 degrees to the ground, the cannon ball will hit the
wall. How high off the ground will the ball hit the wall?
Make a plot of height of cannonball hit as a function
of the angle theta.
Group G problem
Block 1, which has mass m1, slides down a ramp of height h1
and angle theta, as shown in the diagram. Simultaneously,
block 2, of mass m2, slides down another ramp of angle theta
and of height h2. The two blocks started at rest at the top of
their ramps. Somewhere on the frictionless floor between the
two ramps, the two blocks collide and stick together:
- Find the magnitude and direction of their velocity
after the collision and express it in terms of
masses and initial heights of the blocks, g, and
so on.
- Assuming that the ratio of heights of the ramps is
h1/h2 = 2, make two plots of the final velocity
(after the collision) as a function of the mass
ratio m1/m2. One plot covers ratios from 0.1 to 1
and the other should cover 1 to 10.
- Now for fixed mass ratio, m1/m2 = 2, plot the
final velocity as a function of the ratio of
heights of the ramps, h1/h2. Explain why you
believe that all of your plots are consistent.
Group H problem
A spring gun sits on a tabletop that is 1.2 meters above
the floor. If the spring is compressed by 25 mm, it
can fire a 75 gram marble so that it hits 4.2 meters from
the bottom of the table. Neglect all frictional effects.
- Write down a function which give the kinetic energy
of the marble as a function of time during its flight
and plot it.
- Determine the spring constant of the spring.
- How far away from the table will the marble land
if the spring is compressed to 37 mm instead of
25 mm before release? Write down a function which
gives the range of the marble (i.e. distance from
the tabletop) as a function of compression of the
spring and plot it for compression values from
10 mm to 100 mm.
Group I problem
A projectile is fired from level ground at an angle of
theta = 60 degrees from the horizontal. Its initial speed is
v0 = 50 m/s. At the exact midpoint of its trajectory, the
projectile explodes into two equal mass pieces which then follow
different trajectories. Both pieces of the projectile
eventually strike the ground at the same instant.
- Find the time expired from the moment of the
explosion until the moment when both pieces
strike the ground.
- One of the pieces lands a distance d = 155
meters away from the original firing point of
the projectile. Find the distance from the original
firing point for place where the second piece lands.
- Plot the trajectory the projectile would have followed
if it had not exploded and the trajectories of the
two pieces from the explosion on the same graph.