A river flows with a uniform velocity v. A person in a motorboat travels 1.21km upstream, at which time she passes a log floating by. Always with the same throttle setting, the boater continues to travel upstream for another 52.6min and then returns downstream to her starting point, which she reaches just as the same log does. Calculate the velocity of the river.
Ah, trying to figure out the velocity of the river, are we? Well, let's set sail into the world of physics and see what we can come up with!
Now, when the person in the motorboat travels upstream, they pass the log while going against the current. Then, after some time, they turn around and go downstream, eventually reaching their starting point just as the log does. Fascinating!
Let's break it down. For the person in the boat to reach their starting point at the same time as the log, that means their total travel time upstream and downstream must be the same. So, the time it takes for the log to travel downstream from the point where the boater passes it must be equal to the time it takes for the boater to travel upstream.
Now, the distance traveled upstream is given as 1.21 km. Let's convert that to meters, shall we? 1.21 km is equal to 1210 meters. Here comes some math!
Since time equals distance divided by velocity, we can say that the time taken upstream by the boater is 1210/v, where v is the velocity of the river.
To find the time it takes for the log to travel downstream, we'll need to know the total distance traveled by the boater. But worry not! The boater travels the same distance upstream as downstream, so the total distance is just 2 x 1210 meters.
Now, let's focus on that downstream time. The distance covered is 2 x 1210 meters, and the time is given as 52.6 minutes. But we need to convert that to hours, so it becomes 52.6/60 hours.
So, we can say that the time taken downstream by the boater is 2 x 1210/v + 52.6/60.
Since we know that the total travel time upstream and downstream is the same, we can set up an equation:
1210/v = 2 x 1210/v + 52.6/60
Now, solve this equation for v, and voila! You'll have the velocity of the river.
Just remember, my friend, when it comes to solving equations and calculating velocities, take it all with a grain of river salt and a splash of humor!
To calculate the velocity of the river, we can use the concept of relative velocity.
Let's assume:
- The velocity of the river is v_r
- The velocity of the motorboat in still water is v_mb
- The time taken to reach the log while traveling upstream is t
When the person in the motorboat is going upstream towards the log:
- The relative velocity of the motorboat with respect to the log is v_mb - v_r (as they are moving against the current)
- The distance covered by the motorboat to reach the log is 1.21 km (given)
Using the formula: distance = velocity x time,
1.21 km = (v_mb - v_r) x t
When the person in the motorboat is coming downstream back to the starting point:
- The relative velocity of the motorboat with respect to the log is v_mb + v_r (as they are moving with the current)
- The time taken to reach the starting point is 52.6 min (given)
Using the formula: distance = velocity x time,
1.21 km = (v_mb + v_r) x (52.6/60) (converting time from minutes to hours)
Now we have a system of two equations:
1. 1.21 km = (v_mb - v_r) x t
2. 1.21 km = (v_mb + v_r) x (52.6/60)
We can solve this system of equations to find the velocities of the river and the motorboat.
Let's expand the second equation:
1.21 km = (v_mb + v_r) x 0.877 (approx.)
Rearranging the equations, we get:
1.21 km/t = v_mb - v_r
1.21 km/0.877 = v_mb + v_r
Now, we can add the two equations together:
(1.21 km/t) + (1.21 km/0.877) = (v_mb - v_r) + (v_mb + v_r)
Simplifying the equation, we get:
1.21 km/t + 1.38 km ≈ 2v_mb
Dividing both sides of the equation by 2:
(1.21 km/t + 1.38 km) / 2 ≈ v_mb
Now, we can substitute the value of v_mb in the equation 1.21 km = (v_mb - v_r) x t:
1.21 km = ((1.21 km/t + 1.38 km) / 2 - v_r) x t
Let's solve this equation for the velocity of the river (v_r) by isolating v_r:
1.21 km = ((1.21 km/t + 1.38 km) / 2 - v_r) x t
Distribute the t:
1.21 km = (1.21 km/t + 1.38 km)/2 - v_r x t
Multiply both sides by 2:
2.42 km = 1.21 km/t + 1.38 km - 2v_rt
Rearrange the equation:
0 = 1.21 km/t + 1.38 km - 2v_rt - 2.42 km
Now, we can solve this equation for v_r:
2v_rt = 1.21 km/t + 1.38 km - 2.42 km
Rearranging and simplifying:
v_rt = (1.21 km/t + 1.38 km - 2.42 km) / 2
Substitute the value of t (52.6 min = 0.877 hours):
v_r = (1.21 km/0.877 hours + 1.38 km - 2.42 km) / 2
Calculating the values:
v_r = (1.38 km - 1.61 km) / 2
v_r = -0.23 km / 2
Therefore, the velocity of the river is -0.115 km/h.
To solve this problem, we need to use the concept of relative velocity and apply it to the motion of the boater and the log in the river.
Let's break down the problem into two parts: the upstream motion and the downstream motion.
1. Upstream motion:
When the boater is moving upstream, the effective velocity is the difference between the velocity of the boat (v_b) and the velocity of the river (v_r). Therefore, the effective velocity during the upstream motion is (v_b - v_r).
The distance traveled upstream is given as 1.21 km. We can convert it to meters for convenience:
Distance upstream = 1.21 km = 1210 meters.
Now, we need to find the time it takes for the boater to travel 1210 meters upstream. We are given that the time taken is 52.6 minutes, which we need to convert to seconds for consistency:
Time upstream = 52.6 min = 52.6 x 60 = 3156 seconds.
Using the formula: Distance = Velocity x Time, we can write the equation for the upstream motion:
1210 = (v_b - v_r) x 3156.
2. Downstream motion:
When the boater is moving downstream, the effective velocity is the sum of the velocity of the boat (v_b) and the velocity of the river (v_r). Therefore, the effective velocity during the downstream motion is (v_b + v_r).
We are given that the time taken to travel downstream is the same as the time it takes for the log to travel downstream and reach the starting point.
Using the formula: Distance = Velocity x Time, and knowing that the distance covered is 1210 meters, we can write the equation for the downstream motion:
1210 = (v_b + v_r) x Time.
Now, here comes the trick. The boater takes the same length of time to travel upstream and downstream:
3156 seconds = Time.
We have two equations now:
1. 1210 = (v_b - v_r) x 3156.
2. 1210 = (v_b + v_r) x 3156.
To solve these equations, we can use the method of substitution.
We multiply equation 1 by (v_b + v_r) and equation 2 by (v_b - v_r):
1. 1210 x (v_b + v_r) = 3156 x (v_b - v_r).
2. 1210 x (v_b - v_r) = 3156 x (v_b + v_r).
Expanding these equations gives us two equations:
1. 1210v_b + 1210v_r = 3156v_b - 3156v_r.
2. 1210v_b - 1210v_r = 3156v_b + 3156v_r.
Rearranging the terms gives us:
1. 1946v_b = 4366v_r.
2. -1946v_r = 4376v_b.
Now divide both sides of equation 1 by 1946v_r and equation 2 by 4376v_b:
1. v_b/v_r = 4366/1946.
2. -v_r/v_b = 4376/1946.
Since the magnitude of the velocity cannot be negative, we can ignore equation 2 and only focus on equation 1:
v_b/v_r = 4366/1946.
Now, substitute v_b/v_r = 4366/1946 into equation 1 and solve for v_r:
4366/1946 = 4366v_r/1946.
Divide both sides by 4366:
v_r/1946 = 1.
Multiply both sides by 1946:
v_r = 1946 m/s.
Therefore, the velocity of the river is 1946 m/s.
The velocity of the river is equal to the speed of the motorboat divided by the time it took to travel upstream and back.
Speed of the motorboat = 1.21km / (52.6min x 60sec/min) = 0.0045 m/s
Time to travel upstream and back = 52.6min x 2 = 105.2min
Velocity of the river = 0.0045 m/s / (105.2min x 60sec/min) = 0.00008 m/s