A sliotar struck high up the pitch in Croke Park, a parcel falling from an aircraft in flight, a boat crossing the Shannon while the current pushes it along: in all three cases there are two motions happening at once. Galileo was the first to realise something that seems simple but cleared up centuries of confusion: those two motions can be studied separately, as if the other did not exist, and then combined using the vector addition you already know. Use $s$, $u$, $v$, $a$ and $t$ as in the rest of the course.
The principle of independence of motions
A stone dropped from the top of the mast of a moving ship lands exactly at the foot of the mast and not several metres behind it, as the ancient scholars believed: how is it possible that the ship's horizontal motion does not alter its vertical fall at all?
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A train crosses a bridge $19.6\ \text{m}$ high when an object becomes detached from a carriage. Take $g = 9.8\ \text{m/s}^2$. a) Calculate the time it takes for the…
A train crosses a bridge $19.6\ \text{m}$ high when an object becomes detached from a carriage. Take $g = 9.8\ \text{m/s}^2$.
a) Calculate the time it takes for the object to reach the ground.
b) Reason whether this time depends on the horizontal speed of the train.
If you throw a coin horizontally from the edge of a table at the same time as you drop another one straight down, you will hear a single thud on the floor: why do two objects with such different horizontal velocities take exactly the same time to hit the ground?
In an investigation of a horizontal launch (for example a marble leaving a ramp), record the height, the range and the time several times. Repeated readings reveal random error, and a consistent offset in every reading suggests systematic error. Quote results to a sensible number of significant figures.
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An aid aeroplane flies horizontally at $100\ \text{m/s}$ and at an altitude of $500\ \text{m}$ when it drops a package. Take $g \approx 10\ \text{m/s}^2$. Look at the…
An aid aeroplane flies horizontally at $100\ \text{m/s}$ and at an altitude of $500\ \text{m}$ when it drops a package. Take $g \approx 10\ \text{m/s}^2$. Look at the diagram:
a) Calculate the time it takes for the package to fall.
b) Calculate the horizontal distance the package travels before touching the ground.
An aeroplane flies horizontally at $80\ \text{m/s}$ and at an altitude of $720\ \text{m}$ and must drop a package so that it lands on a specific point on the ground.…
An aeroplane flies horizontally at $80\ \text{m/s}$ and at an altitude of $720\ \text{m}$ and must drop a package so that it lands on a specific point on the ground. Take $g \approx 10\ \text{m/s}^2$.
a) Calculate the fall time of the package.
b) Calculate the horizontal distance travelled by the package during the fall.
c) Reason at what horizontal distance before the target the package should be dropped.
From the top of a cliff $45\ \text{m}$ above the sea, a stone is dropped and, at the same instant, another identical stone is thrown horizontally at $10\ \text{m/s}$.…
From the top of a cliff $45\ \text{m}$ above the sea, a stone is dropped and, at the same instant, another identical stone is thrown horizontally at $10\ \text{m/s}$. Take $g \approx 10\ \text{m/s}^2$. Look at the cliff diagram:
a) Calculate the time it takes for the first stone to fall.
b) Reason, using the principle of independence of motion, which of the two hits the water first.
c) Calculate the horizontal range of the second stone.
d) Calculate the magnitude of the final velocity of each stone and compare the results.
In a laboratory practical, a ball rolls along a table $1.25\ \text{m}$ high and leaves the edge horizontally. A cup is placed on the floor with its centre…
In a laboratory practical, a ball rolls along a table $1.25\ \text{m}$ high and leaves the edge horizontally. A cup is placed on the floor with its centre $1.5\ \text{m}$ from the vertical line of the table edge. Take $g \approx 10\ \text{m/s}^2$. Look at the diagram:
a) Calculate the time taken for the ball to fall from the edge to the ground.
b) Calculate the horizontal velocity with which it must leave the table to land in the glass.
c) Using this velocity, calculate the components and the magnitude of the velocity of the ball upon reaching the ground.
The arc of a sliotar struck from the ground, or a jet of water from a tilted garden hose, traces a symmetrical arc in the air, whose height and range change when you alter the angle: which combination of elementary motions gives rise to that curved path?
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The vertical component of the initial velocity of a projectile is $20\ \text{m/s}$. Take $g \approx 10\ \text{m/s}^2$. a) Calculate the time it takes to reach the…
The vertical component of the initial velocity of a projectile is $20\ \text{m/s}$. Take $g \approx 10\ \text{m/s}^2$.
a) Calculate the time it takes to reach the highest point of its trajectory.
b) If it launches from the ground and returns to the ground, how long is its total flight? Justify your answer.
An arrow leaves the bow with a vertical velocity component of $30\ \text{m/s}$. Take $g \approx 10\ \text{m/s}^2$ and remember that at the highest point the vertical…
An arrow leaves the bow with a vertical velocity component of $30\ \text{m/s}$. Take $g \approx 10\ \text{m/s}^2$ and remember that at the highest point the vertical velocity is zero.
a) Calculate the time it takes to ascend.
b) Calculate the maximum height it reaches above the launch point.
Composition of rectilinear motions: boats and currents
With $v_b=4\ \mathrm{m\,s^{-1}}$ at $90^\circ$ (the boat, perpendicular to the bank) and $v_c=1.5\ \mathrm{m\,s^{-1}}$ at $0^\circ$ (the current, parallel to the bank), the resultant $v$ is the real velocity of the boat relative to the ground: it moves diagonally, combining the crossing with the drift.
Back to the start: the Shannon crossing is just two independent motions, one across and one along the river, combined by Pythagoras.
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A boat crosses a river $40\ \text{m}$ wide at its own velocity of $2\ \text{m/s}$, directed perpendicular to the current. Look at the crossing diagram: a) State…
A boat crosses a river $40\ \text{m}$ wide at its own velocity of $2\ \text{m/s}$, directed perpendicular to the current. Look at the crossing diagram:
a) State Galileo's principle of independence of motion.
b) Apply it to calculate the time the boat takes to cross the river, reasoning why the current does not alter this time.