Anthony is trying to cross the Mississippi river in a kayak. He is traveling due east at 6 ft/second. However there is a current that is traveling due north at 4 ft/sec. find the new speed and direction of Anthony and his kayak. If the distance across the Mississippi is 120 ft, determine if Anthony would make it to the other side faster if there was a current of if there wasn't
new speed = √(6^2+4^2) = √52 ft/s
If he rows for the opposite bank, his crossing takes the same time, regardless of the currents' speed.
To find the new speed and direction of Anthony and his kayak, we can use vector addition. We'll consider east as the positive x-direction and north as the positive y-direction.
Given:
- Anthony's speed due east = 6 ft/s
- Current speed due north = 4 ft/s
We can represent Anthony's velocity as a vector in the x-direction (Vx) and the y-direction (Vy).
Anthony's velocity vector (V) can be expressed as V = Vx + Vy.
Vx = 6 ft/s (eastward)
Vy = 4 ft/s (northward)
Now, we can calculate the magnitude and direction of Anthony's new velocity. The magnitude (Vnew) can be determined using the Pythagorean theorem:
Vnew = √(Vx^2 + Vy^2)
= √(6^2 + 4^2)
= √(36 + 16)
= √52
= 2√13 ft/s
To find the direction θ of Anthony's new velocity, we can use the inverse tangent function (tan^(-1)):
θ = tan^(-1)(Vy / Vx)
= tan^(-1)(4 / 6)
= tan^(-1)(2/3)
≈ 33.69° (with respect to the positive x-axis)
Therefore, Anthony's new speed is approximately 2√13 ft/s at an angle of approximately 33.69° with respect to the positive x-axis.
To determine if Anthony would make it to the other side faster with or without the current, we need to compare the time taken to cross the river in both scenarios.
Without the current:
Distance = 120 ft
Speed = 6 ft/s
Time taken to cross the river without the current = Distance / Speed
= 120 ft / 6 ft/s
= 20 seconds
With the current:
Distance = 120 ft
Speed = 2√13 ft/s
Time taken to cross the river with the current = Distance / Speed
= 120 ft / (2√13 ft/s)
= 60 / √13 seconds
To compare the times, we need to simplify the expression:
Time taken with the current = (60 / √13) * (√13 / √13)
= 60√13 / 13 seconds
Comparing the times, we find that:
Time taken without current = 20 seconds
Time taken with current = 60√13 / 13 seconds
Since 60√13 / 13 is greater than 20, it means that Anthony would take longer to cross the river with the current. Therefore, Anthony would make it to the other side faster without the current.