Exercises: Motion problems

A cheetah runs at a constant speed of $108$ km/h chasing its prey. How many seconds will it take to cover a distance of $300$ m?

An AVE high-speed train travels at an average speed of $280$ km/h. If the journey from origin to destination lasts exactly $2$ hours and $15$ minutes, what total distance has the train covered?

Two cyclists set off at the same time from two towns $90$ km apart, travelling along the same road towards each other. The first cyclist travels at $20$ km/h and the second at $25$ km/h. How long will it take for them to meet?

Two railway stations, A and B, are connected by a straight track of $450$ km. A train leaves A heading for B at $120$ km/h and, at the same time, another train leaves B heading for A at $80$ km/h. How far from station A will they meet?

A car leaves a town at a constant speed of $90$ km/h. Half an hour later, a motorcycle sets off in pursuit along the same road at $120$ km/h. How long will it take for the motorcycle to catch up with the car?

A delivery driver leaves the warehouse driving at $100$ km/h. Realising that they have forgotten an urgent package, a colleague sets off in pursuit from the same warehouse $15$ minutes later, travelling at $120$ km/h. How far from the warehouse will the colleague catch up with them?

Two aeroplanes set off from the same airport at the same time, but fly in opposite directions. One flies at $800$ km/h and the other at $950$ km/h. How far apart will they be after $2,5$ hours of flight?

A delivery van covers $150$ km in $2$ hours, takes a mandatory break of $30$ minutes, and then covers another $120$ km in $1,5$ hours. What was its average speed for the entire journey (including the break)?

Lucía makes a round trip by car between Sevilla and Córdoba along the A-4 motorway, covering a distance of $140\text{ km}$ each way. On the way there, due to thick fog and slow traffic, she maintains an average speed of $70\text{ km/h}$. On the return journey, with the road completely clear, she drives at a constant average speed of $140\text{ km/h}$. To calculate her average speed over the entire trip ($280\text{ km}$), Lucía does the following calculation: $\frac{70 + 140}{2} = 105\text{ km/h}$.

1. a) Explain with reasoning why Lucía's calculation is incorrect without merely stating that "that's not the formula" (think about how much time she spends at each of the two speeds during the journey).
2. b) Calculate the exact time she took on the journey there and on the return journey, and obtain from them the true average speed of the entire trip.
3. c) Why is the actual average speed closer to $70\text{ km/h}$ than to $140\text{ km/h}$?

You are going to cross Madrid using the BiciMAD public electric bicycle service from Plaza de Castilla to Atocha station along the Castellana-Recoletos-Prado axis, which covers an approximate distance of $6\text{ km}$. The bicycles have pedal assistance that allows maintaining an average speed of $15\text{ km/h}$.

1. a) Before carrying out any calculations with formulas, make an intuitive estimate: will it take you more or less than half an hour to complete the route? Justify your deduction by mentally relating the $6\text{ km}$ to the distance you would cover in $60\text{ minutes}$.
2. b) Apply the uniform motion formula to calculate the exact time in hours and convert it to minutes.
3. c) Check whether the exact result agrees with your initial estimate and assess whether this time obtained is competitive compared to making the same journey by city bus during rush hour.

A delivery lorry leaves Madrid for Barcelona at 08:00 at a constant speed of $90$ km/h. At 09:30, a van leaves Barcelona for Madrid at $150$ km/h. If the road distance between the two cities is $615$ km, at what exact time will they pass each other?

A marathon runner moves at a constant pace of $15$ km/h. A support cyclist travelling in the same direction at $25$ km/h is $4$ km behind the runner. In how many minutes will the cyclist catch up with the runner?

A motorboat has a speed of $22$ km/h in still water. If it travels along a river with a current flowing at $4$ km/h, how much total time will it take to make a continuous round trip between two ports separated by $117$ km?

A hiker climbs to the top of a mountain at a speed of $4$ km/h and descends along the same path at $6$ km/h. If the hike took a total of $5$ hours of walking (excluding breaks at the summit), what is the distance from the base to the summit?

During a heavy summer thunderstorm, you see a flash of lightning and $8$ seconds later you hear the thunder. Knowing that the speed of light is approximately $300\,000$ km/s and that of sound in air is $340$ m/s, at approximately what distance in kilometres did the lightning strike occur?

Two ships set sail from the same port at the same time. Ship A sails in a straight line north at a speed of $18$ knots, and ship B sails east at $24$ knots. What distance, in nautical miles, will separate them after a $3$-hour voyage? (Note: $1$ knot is equivalent to $1$ nautical mile per hour).

A driver makes an intercity journey of $300$ km. They drive the first half of the journey at a constant speed of $120$ km/h on a motorway, and the second half at $80$ km/h on a main road. What was their average speed for the entire journey?

A high-speed train measuring $200$ m in length travels at $252$ km/h. How many seconds will it take to pass completely through a tunnel $1,2$ km long? (It is considered to have passed through the tunnel from the moment the engine enters to the moment the rear carriage leaves it).

Two cycle tourists, Mario and Rocío, travel along the Vía Verde de la Sierra (Cádiz-Sevilla), which has a total length of $36\text{ km}$. At $10:00\text{ h}$, they set off at the same time towards each other from opposite ends of the route. Mario pedals at a constant speed of $11\text{ km/h}$ and Rocío at $13\text{ km/h}$. To calculate at what time and at what distance from Mario's starting point they will cross paths, two methods are proposed:

• Method A (Arithmetic / approach speed): Add their speeds together to calculate the combined rate at which the distance between them decreases ($11 + 13 = 24\text{ km/h}$) and divide the total distance by that sum.

• Method B (Algebraic): Set up a linear equation by adding the distances covered by each of them and setting it equal to the total distance: $e_1 + e_2 = 36$, that is, $11 \cdot t + 13 \cdot t = 36$.

1. a) Solve the problem applying both methods and verify that you reach the same exact meeting time.
2. b) Explain why both methods are mathematically equivalent.
3. c) Imagine that Rocío gets a flat tyre before setting off and leaves $30\text{ minutes}$ later than Mario. Justify which of the two methods is safer and more direct to adapt the calculation to this new situation without getting confused by the data.

During the traditional Descenso del Sella (Asturias), the river water flows continuously at a current speed of $3\text{ km/h}$. A canoeist carries out a training session along a $12\text{ km}$ stretch between Arriondas and an intermediate finish line.

1. a) Paddling downstream (with the current), she completes the $12\text{ km}$ section in exactly $1\text{ hour}$. Model the effective downstream speed in terms of the athlete's own paddling speed ($v_p$) and calculate at what speed she would move if she paddled in a calm water reservoir.
2. b) If she then decides to return upstream (against the current) covering the same $12\text{ km}$ at that same paddling pace of her own, calculate how long it will take her to complete the return.
3. c) If, due to heavy rains, the current speed were to temporarily increase to $9\text{ km/h}$, what would happen physically when trying to paddle back upstream at her usual pace? Justify your answer based on the data.

On a circular running track $400$ m long, two runners set off from the same point at the same time. Runner A runs at $6$ m/s and runner B at $8$ m/s. Calculate how long it will take for them to meet for the first time if they run in opposite directions, and how long it will take for them to meet if they run in the same direction.

A professional rower takes $2$ hours to row $12$ km upstream against the current and $1$ hour to row the same distance downstream with the current. Set up a system of equations to determine the speed of the boat in still water and the speed of the river current.

Vehicle A sets off from a town at a speed of $v$ km/h. A time of $h$ hours later, vehicle B sets off in pursuit from the same point at a speed of $v + k$ km/h. Express algebraically the time it takes vehicle B to catch up with vehicle A in terms of the parameters $v$, $h$ and $k$.

Two towns A and B are $300$ km apart. A car leaves A heading for B at $100$ km/h and another car leaves B heading for A at $80$ km/h at exactly the same time. At what two times will the distance separating them be exactly $30$ km?

At an airport, a horizontal moving walkway $120$ m long moves at $1$ m/s. A person walks along the walkway at a constant speed $v$ relative to the walkway itself. If it takes them $40$ seconds to travel along the walkway walking in the direction of motion, how long would it take them walking against the direction of motion of the walkway?

A parade $3$ km long moves along an avenue at a constant speed of $5$ km/h. A photographer on a bicycle sets off from the back of the procession, rides to the front to take some photos, and immediately returns to the back. If the speed of the bicycle is $15$ km/h, what total distance has the photographer covered relative to the ground?

In a $100$ km mountain relay race, the first runner runs at $12$ km/h and hands over the baton to the second runner, who runs at $16$ km/h to reach the finish line. If the total time of the race was $7$ continuous hours, what distance in kilometres did each of the two runners cover?

Two moving objects travel at constant speed around a track. If they move in the same direction, the faster one catches up with the slower one every $20$ seconds. If they move in opposite directions, they pass each other every $4$ seconds. Knowing that the track is circular and $100$ m long, calculate the speed in m/s of each moving object.

In the Adif technical monitoring room, the crossing of two trains travelling along a $450\text{ km}$ railway line between Zaragoza and Madrid is being monitored. The computer control system sets up the following equation to determine the time of the crossing:

$v \cdot 2 + (v + 25) \cdot 2 = 450$

1. a) Interpret each of the elements of the equation mathematically and write a coherent, complete problem statement that leads exactly to that expression (you must clearly define what quantity the unknown $v$ represents, what the term $(v + 25)$ means, what the number $2$ represents, and what the sum equated to $450$ means).
2. b) Solve the equation, calculate the average speed of each of the two trains, and find at what distance from Zaragoza the meeting occurs if the train travelling at speed $v$ departed from that city.
3. c) Modify the original equation to represent a different situation: both trains maintain the calculated speeds, but the second train is delayed and departs $20\text{ minutes}$ after the first.

The Directorate-General for Traffic (DGT) has an average speed camera installed on the AP-6 motorway in the Sierra de Guadarrama. The monitored section measures exactly $8\text{ km}$ and the maximum legal speed limit is $100\text{ km/h}$. A vehicle enters this section at $15:00:00\text{ h}$ and covers the first $5\text{ km}$ at a constant speed of $125\text{ km/h}$. At that point, the driver notices the speed trap and immediately reduces his speed, keeping it constant for the remaining $3\text{ km}$.

1. a) Determine the minimum time required by law to cover the full $8\text{ km}$ of the section without being fined (express it in minutes and seconds) and calculate how much time the driver has already spent on the first $5\text{ km}$.
2. b) Model and calculate the maximum constant speed at which he can cover the remaining $3\text{ km}$ so that the average speed over the entire section does not exceed $100\text{ km/h}$. Would it be enough for him to reduce his speed to exactly $100\text{ km/h}$? Justify your answer.
3. c) If during those last $3\text{ km}$ he drives at $75\text{ km/h}$, at what exact time (hours, minutes, and seconds) will the exit camera record his passage? Conclude, by comparing the times, whether or not he will receive a fine.