A man can swim straight across a 100m wide river in 4 min if there is no current. If there is current it takes 5 min to cross straight. Determine the velocity of the current.

To determine the velocity of the current, we can use the information provided about the man's swimming ability with and without the current. 

### Step-by-Step Solution:

1. **Without Current:**
   - The man swims straight across the river, which is 100 meters wide, in 4 minutes.
   - His speed without the current can be calculated as:
   \[
   v_{\text{man}} = \frac{100 \text{ meters}}{4 \text{ minutes}} = 25 \text{ m/min}.
   \]

2. **With Current:**
   - When there is a current, it takes him 5 minutes to cross the same distance straight.
   - Since he's able to swim straight across, this means that his swimming speed relative to the riverbank must be slower due to the current affecting his motion. The time increases from 4 minutes to 5 minutes.

3. **Effect of the Current:**
   - The current acts perpendicular to his swimming direction, and he must compensate for this to swim directly across the river.
   - Let \( v_{\text{current}} \) be the velocity of the current.
   - The man’s velocity with respect to the riverbank forms a right triangle, where:
     - The hypotenuse is the man’s swimming speed relative to the water (25 m/min),
     - The base (horizontal side) is the velocity across the river (which is \( \frac{100 \text{ meters}}{5 \text{ minutes}} = 20 \text{ m/min} \)),
     - The perpendicular side is the velocity of the current, \( v_{\text{current}} \).

   Using the Pythagorean theorem:
   \[
   v_{\text{man}}^2 = v_{\text{across}}^2 + v_{\text{current}}^2,
   \]
   \[
   25^2 = 20^2 + v_{\text{current}}^2,
   \]
   \[
   625 = 400 + v_{\text{current}}^2,
   \]
   \[
   v_{\text{current}}^2 = 625 - 400 = 225,
   \]
   \[
   v_{\text{current}} = \sqrt{225} = 15 \text{ m/min}.
   \]

### Final Answer:
The velocity of the current is **15 meters per minute**.

Would you like further details or have any questions?

Here are 5 related questions to extend your understanding:
1. What would the man's effective speed be if the current helped him swim across?
2. How would the time change if the current’s velocity doubled?
3. What if the river were 150 meters wide but the same current existed?
4. How does the Pythagorean theorem help in resolving this problem?
5. What happens if the current flows at the same speed as the man's swimming speed?

**Tip:** Always break down motion problems into components and use trigonometric relations or the Pythagorean theorem for perpendicular velocities.

Learn how to determine the velocity of a river's current using a swimmer's crossing times. This problem uses kinematic concepts and the Pythagorean theorem to calculate the current's speed, suitable for students in Grades 9-12. Follow the step-by-step solution to understand the principles of relative velocity.

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