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 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 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.

A tennis ball leaves the racket at $v_0 = 20\ \text{m/s}$ at an angle of $60^\circ$ to the horizontal. Use $\sen 60^\circ \approx 0.87$ and $\cos 60^\circ = 0.5$.

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

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.

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 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.

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.

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.

A handball goalkeeper throws the ball at $v_0 = 20\ \text{m/s}$ at an angle of $30^\circ$ to the horizontal. Use $\sen 30^\circ = 0.5$, $\cos 30^\circ \approx 0.87$ and $g \approx 10\ \text{m/s}^2$.

1. a) Calculate the components of the initial velocity.
2. b) Calculate the flight time until the ball returns to the launch height.
3. c) Calculate the range.
4. d) Calculate the maximum height.

A volleyball player serves at $v_0 = 25\ \text{m/s}$ at an angle of $53^\circ$ to the horizontal. Use $\sen 53^\circ \approx 0.8$, $\cos 53^\circ \approx 0.6$ and $g \approx 10\ \text{m/s}^2$.

1. a) Calculate the components of the initial velocity.
2. b) Reason what the vertical component of the velocity is at the highest point and calculate the speed at that point.
3. c) Calculate the maximum height and the time taken to reach it.

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 rugby player kicks the ball from the ground at $v_0 = 30\ \text{m/s}$ at an angle of $53^\circ$ to the horizontal. Use $\sen 53^\circ \approx 0.8$, $\cos 53^\circ \approx 0.6$ and $g \approx 10\ \text{m/s}^2$. The figure shows the trajectory of the ball and the posts.

1. a) Calculate the components of the initial velocity.
2. b) If the posts are $36\ \text{m}$ from the point of the kick, calculate at what height the ball passes over the goal line and determine whether it clears the crossbar, located at a height of $3\ \text{m}$.
3. c) Calculate the maximum height reached by the ball.
4. d) Calculate the range of the ball and determine whether it would clear the dead-ball line, located $80\ \text{m}$ from the point of the kick.

For a launch over level ground, the range is given by $x = \frac{v_0^2 \times \sen(2\theta)}{g}$, where $\theta$ is the launch angle.

1. a) Using this expression, justify that the range is maximum when $\theta = 45^\circ$.
2. b) A javelin thrower wants to reach $80\ \text{m}$ throwing at $45^\circ$; take $g \approx 10\ \text{m/s}^2$ and calculate the required initial velocity.
3. c) With this same initial velocity, calculate the range if thrown at $30^\circ$ and check whether it reaches $80\ \text{m}$. Use $\sen 60^\circ \approx 0.87$.

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.

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.

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.

An aeroplane flies east at its own velocity of $200\ \text{m/s}$ while the wind blows constantly northwards at $40\ \text{m/s}$.

1. a) Calculate the magnitude of the aeroplane's resultant velocity relative to the ground.
2. b) Calculate the drift angle: the angle by which its path deviates from the desired easterly course.
3. c) If the flight lasts $500\ \text{s}$, calculate the position of the aeroplane (distance travelled east and north of the starting point) and the distance from the point it would have reached without wind.

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 center $1.5\ \text{m}$ 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.

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.