Moving-object problems
Two cyclists set off at the same time, one from each end of a road, and pedal towards each other. How long do they take to meet? A car sets off with a half-hour head start and a faster motorbike sets off behind it. Does it catch up, and where? All these questions are solved with the same idea: $e = v \cdot t$, and a good drawing.
The relationship between distance, speed and time
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?
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…
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?
Meeting: two objects approaching each other
The position–time graph shows this well: two lines going in opposite directions that cross exactly at the moment and place where they meet.
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…
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…
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 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…
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?
Chasing or catching up
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…
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…
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?
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…
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?
Average speed over journeys with different stages
Include stops or pauses in the same way: they count as time at zero speed, and they lengthen the total time without adding distance.
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…
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)?
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 $80$…
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?
Currents and belts: speeds that add or subtract
When the medium is also moving (a river, a conveyor belt), the real speed of the moving object is its own speed plus or minus that of the medium, depending on whether it goes with or against it.
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…
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 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…
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.
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…
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?
Back to the start: the motorbike from the beginning does catch up with the car: it takes $1{,}5\text{ h}$ to do so and catches it $180\text{ km}$ from the starting point ($120 \cdot 1{,}5 = 180\text{ km}$).