Question

You swim at 3 km/h with your body perpendicular to a stream with a current of 5 km/h. What is your actual velocity, considering the vector sum of the stream's velocity and your swimming velocity?

a) (2 , textkm/h)
b) (4 , textkm/h)
c) (5 , textkm/h)
d) (8 , textkm/h)

Answer 1

Your actual velocity when swimming perpendicular at 3 km/h to a stream with a 5 km/h current is approximately 5.83 km/h, found using vector addition and the Pythagorean theorem. The closest multiple-choice option is (c) 5 km/h, although this doesn't exactly match the calculated result.

Step-by-step explanation:

To determine your actual velocity when swimming across a stream with a current, you need to use vector addition. In this scenario, your swimming velocity is perpendicular to the current's velocity. If you swim at a speed of 3 km/h and the stream's current is 5 km/h, we can use the Pythagorean theorem to find the resultant velocity since the two velocities are at right angles to each other.

Using the Pythagorean theorem, your actual velocity (Vtotal) can be calculated as follows:

Vtotal = \( \sqrt{V_{swim}^2 + V_{current}^2} \) = \( \sqrt{3^2 + 5^2} \) = \( \sqrt{9 + 25} \) = \( \sqrt{34} \)

Therefore, Vtotal is approximately 5.83 km/h, which is closest to option (c) in your multiple-choice answers.

Answer 2

To find the actual velocity, we need to find the vector sum of the swimming velocity (3 km/h) and the stream's velocity (5 km/h). The resultant velocity is approximately 5.83 km/h at an angle of 30.96° with the stream's direction. Thus, the correct answer is c) (5 , textkm/h).

Step-by-step explanation:

To find the actual velocity, we need to find the vector sum of the swimming velocity (3 km/h) and the stream's velocity (5 km/h). Since the directions are perpendicular, we can use the Pythagorean theorem to find the magnitude of the resultant velocity:

Let Vs be the swimming velocity and Vc be the current velocity.

The magnitude of the actual velocity (V) is given by: V= √Vs²+Vc²

In this case:

Swimming velocity (Vs) = 3 km/h

Stream velocity (Vc) = 5 km/h

V= √3²+5²

V= √9+25

V= √34

Therefore, the actual velocity is approximately 5.83 km/h.

However, this question seems to be asking for a vector in the form of (magnitude, direction).

To find the direction, you can use the tangent inverse:

θ=tan⁻¹ (Vs/Vc)

​θ=tan⁻¹ (3/5)

θ≈30.96°

So, the actual velocity is approximately (5.83, 30.96) km/h.

None of the provided options exactly match this, but the closest option is (5, textkm/h).

So, the correct answer is c) (5 , textkm/h).