Given that a man's speed with the current (downstream) is 15 km/h and the speed of the current is 2.5 km/h, his speed while going against the current (upstream) is 10 km/h. This calculation involves understanding the principles of relative speed in water.
Understanding Relative Speed in Water
When an object moves in water, its effective speed is influenced by the speed of the water itself (the current). This concept is fundamental in solving problems related to boats and streams.
- Speed in Still Water (u): This is the actual speed of the man or boat without any influence from the current.
- Speed of the Current (v): This is the speed at which the water is flowing.
- Speed Downstream (with current): When moving in the same direction as the current, the speeds add up.
- Formula: Downstream Speed = u + v
- Speed Upstream (against current): When moving in the opposite direction to the current, the current's speed is subtracted from the object's speed in still water.
- Formula: Upstream Speed = u - v
Calculating the Speed Against the Current
To find the man's speed against the current, we first need to determine his speed in still water using the information provided.
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Identify Given Information:
- Speed with current (Downstream Speed): 15 km/h (u + v = 15)
- Speed of the current: 2.5 km/h (v = 2.5)
-
Calculate Speed in Still Water (u):
We know that Downstream Speed = Speed in Still Water + Speed of Current.
So, u + v = 15 km/h.
Substituting the value of v:
u + 2.5 km/h = 15 km/h
u = 15 km/h - 2.5 km/h
u = 12.5 km/h (This is the man's speed in still water). -
Calculate Speed Against the Current (Upstream Speed):
Now that we have the man's speed in still water (u) and the speed of the current (v), we can find his speed against the current.
Upstream Speed = Speed in Still Water - Speed of Current.
Upstream Speed = u - v
Upstream Speed = 12.5 km/h - 2.5 km/h
Upstream Speed = 10 km/h
Therefore, the speed of the man while going against the current is 10 km/h.
Summary of Speeds
| Type of Speed | Formula (u = speed in still water, v = speed of current) | Value (km/h) |
|---|---|---|
| Speed with Current | u + v | 15 |
| Speed of Current | v | 2.5 |
| Speed in Still Water | u = (Downstream Speed - v) | 12.5 |
| Speed Against Current | u - v | 10 |
Practical Applications of Relative Speed
Understanding how speeds combine and subtract is crucial in various real-world scenarios, not just for boats in rivers.
- Aviation: Calculating the effective speed of an aircraft taking into account wind speed and direction (tailwind or headwind).
- Conveyor Belts/Escalators: Determining the time it takes for a person to move on a moving surface.
- Fluid Dynamics: Analyzing the movement of objects within fluid flows.
These calculations help in planning travel, ensuring safety, and optimizing performance across different fields. For further reading on relative motion, you can explore resources like Khan Academy's lessons on relative velocity.
