Projectile Motion and Composition of 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 drone flies north at $4\ \text{m/s}$ while the wind pushes it east at $3\ \text{m/s}$.

1. a) Calculate the magnitude of the drone's resultant velocity relative to the ground.
2. b) Taking into account the independence of motion, calculate how long it takes to travel $100\ \text{m}$ northwards.

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.

A helicopter flies horizontally at $30\ \text{m/s}$ and at an altitude of $20\ \text{m}$ when it drops a package. Take $g \approx 10\ \text{m/s}^2$.

1. a) Calculate the fall time of the package.
2. b) Calculate the horizontal range of the package.
3. c) Calculate the velocity components on hitting the ground and its magnitude.

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.

A ski jumper leaves the ramp with a horizontal velocity of $20\ \text{m/s}$ from a height of $80\ \text{m}$ above the landing area. Take $g \approx 10\ \text{m/s}^2$.

1. a) Calculate the flight time and the horizontal range.
2. b) Calculate the velocity components on reaching the landing area and its magnitude.
3. c) Calculate the angle that the final velocity makes with the horizontal.

A ball is kicked with an initial velocity $v_0 = 50\ \text{m/s}$ at an angle of $37^\circ$ to the horizontal. Use $\sen 37^\circ \approx 0.6$ and $\cos 37^\circ \approx 0.8$. The figure shows the resolution of the initial velocity.

1. a) Calculate the horizontal component of the initial velocity.
2. b) Calculate the vertical component of the initial velocity.

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.

A golf ball leaves the tee at $v_0 = 50\ \text{m/s}$ at an angle of $37^\circ$ to the horizontal. Use $\sen 37^\circ \approx 0.6$, $\cos 37^\circ \approx 0.8$ and $g \approx 10\ \text{m/s}^2$. The trajectory of the shot is shown in the figure.

1. a) Calculate the components of the initial velocity.
2. b) Calculate the flight time.
3. c) Calculate the range over level ground.
4. d) Calculate the maximum height.

In a historical reenactment, a cannon fires projectiles at $v_0 = 40\ \text{m/s}$. Take $g \approx 10\ \text{m/s}^2$, $\sen 37^\circ \approx 0.6$, $\cos 37^\circ \approx 0.8$ and recall that $\sen(2\theta) = 2 \times \sen\theta \times \cos\theta$.

1. a) Calculate the range for a firing angle of $37^\circ$.
2. b) Calculate the range for the complementary angle, $53^\circ$, and compare the two results.
3. c) Calculate the maximum height and the flight time in each of the two cases.
4. d) Explain what property pairs of complementary angles have in projectile motion, and what is gained and lost by choosing the higher angle.

An archer shoots an arrow at $v_0 = 25\ \text{m/s}$ at an angle of $37^\circ$ to the horizontal. Use $\sen 37^\circ \approx 0.6$, $\cos 37^\circ \approx 0.8$ and $g \approx 10\ \text{m/s}^2$.

1. a) Calculate the components of the initial velocity.
2. b) Write the equations $x(t)$ and $y(t)$, eliminate time and obtain the equation of the trajectory $y(x)$.
3. c) There is a billboard $5\ \text{m}$ high at a distance of $20\ \text{m}$ from the launch point: calculate the height at which the arrow passes over it and determine whether it clears it.
4. d) Calculate the range of the arrow on flat horizontal ground.

A boat with its own speed of $4\ \text{m/s}$ crosses a river $80\ \text{m}$ wide, pointing its bow perpendicular to the current, which flows at $1.5\ \text{m/s}$. Look at the crossing diagram:

1. a) Calculate the crossing time.
2. b) Calculate the drift: the distance the current has carried it downstream on reaching the opposite bank.
3. c) Calculate the magnitude of the total displacement between the starting point and the arrival point.

A boat skipper wants to cross a river $120\ \text{m}$ wide, arriving directly opposite the starting point. Their boat reaches $5\ \text{m/s}$ in still water and the current flows downstream at $3\ \text{m/s}$. The velocity diagram is shown in the figure.

1. a) Calculate the upstream angle at which the bow must be directed, knowing that it satisfies $\sen\theta = \frac{3}{5}$.
2. b) Calculate the magnitude of the boat's resultant velocity relative to the banks.
3. c) Calculate the time it takes to cross.

A boat capable of travelling at $5\ \text{m/s}$ in still water must cross a river $100\ \text{m}$ wide with a current of $3\ \text{m/s}$.

1. a) If it points its bow perpendicular to the bank, calculate the crossing time, the drift and the magnitude of the total displacement.
2. b) If it wants to arrive directly opposite, calculate the upstream angle it must take (satisfying $\sen\theta = \frac{3}{5}$), the resultant velocity and the crossing time.
3. c) Compare both strategies and reason which minimises the crossing time and which minimises the length of the path relative to the bank, supporting your answer with the results obtained.