Exercises: Compound motions

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:

1. a) State Galileo's principle of independence of motion.
2. b) Apply it to calculate the time the boat takes to cross the river, reasoning why the current does not alter this time.

A train crosses a bridge $19.6\ \text{m}$ high when an object becomes detached from a carriage. Take $g = 9.8\ \text{m/s}^2$.

1. a) Calculate the time it takes for the object to reach the ground.
2. b) Reason whether this time depends on the horizontal speed of the train.

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:

1. a) Calculate the time it takes for the package to fall.
2. b) Calculate the horizontal distance the package travels before touching the ground.

The vertical component of the initial velocity of a projectile is $20\ \text{m/s}$. Take $g \approx 10\ \text{m/s}^2$.

1. a) Calculate the time it takes to reach the highest point of its trajectory.
2. 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 velocity is zero.

1. a) Calculate the time it takes to ascend.
2. b) Calculate the maximum height it reaches above the launch point.

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 $g \approx 10\ \text{m/s}^2$.

1. a) Calculate the fall time of the package.
2. b) Calculate the horizontal distance travelled by the package during the fall.
3. 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 $10\ \text{m/s}$. Take $g \approx 10\ \text{m/s}^2$. Look at the cliff diagram:

1. a) Calculate the time it takes for the first stone to fall.
2. b) Reason, using the principle of independence of motion, which of the two hits the water first.
3. c) Calculate the horizontal range of the second stone.
4. 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 $1.5\ \text{m}$ from the vertical line of the table edge. Take $g \approx 10\ \text{m/s}^2$. Look at the diagram:

1. a) Calculate the time taken for the ball to fall from the edge to the ground.
2. b) Calculate the horizontal velocity with which it must leave the table to land in the glass.
3. c) Using this velocity, calculate the components and the magnitude of the velocity of the ball upon reaching the ground.

Unit challenge. Investigate the kinematics of a horizontal projectile launch by analyzing the influence of the environment.

1. Thermodynamic variables: Check Met Éireann's website for the current temperature $T$ ($^\circ\text{C}$) and pressure $P$ ($\text{hPa}$) in your locality to record the thermodynamic state of the surrounding air.

2. Rectilinear and circular motions: Use an inclined ramp (UARM) or an inverted bicycle wheel rotating uniformly (UCM) with a known radius, for example $R = (30.0 \pm 0.1)\text{ cm}$, to launch a marble with an initial horizontal velocity $v_0$. Determine $v_0$ independently of the range: for the wheel, measure $\omega$ by timing 10 revolutions and use $v_0 = \omega R$; for the ramp, time the marble over a final horizontal section of known length. Estimate the uncertainty of each $v_0$.

3. Composite motion and measurements: Launch the marble horizontally from the edge of a table located at a height $h = (80.0 \pm 0.5)\text{ cm}$. Measure the horizontal range $x$ for 5 different initial velocities using a tape measure (uncertainty $\pm 0.1\text{ cm}$), performing 3 trials for each to calculate the mean value and dispersion.

4. Final product: Prepare a brief report containing the experimental data table, a plot of the range $x$ versus $v_0$, and a reasoned discussion on whether the values of the ambient thermodynamic variables cause noticeable deviations compared to ideal projectile motion in a vacuum.

Evaluation criteria:

• Rigorous collection of experimental measurements and proper expression of their uncertainties.

• Technical quality in the preparation of the table and the kinematic graph.

• Coherent integration of the types of motion (rectilinear, circular, and composite) with thermodynamic variables.

• Critical thinking and clear presentation in the report's conclusions.