two canoeists start paddling at the same time and head toward a small island in a lake. caneist 1 paddles with a speed of 1.35 m/s at an angle of 45 degrees north of east. canoeist 2 starts on the opposite shore of the lake,a distance of 1.5 km due east of canoeist 1. what speed must canoeist 2 have if the two canoes are to arrive at the island at the same time?
To solve this problem, we first need to calculate the time it takes for canoeist 1 to reach the island. Then we can use this time to find the required speed for canoeist 2.
Let's start by finding the time it takes for canoeist 1 to reach the island. We can break down its velocity into its x and y components:
Vx1 = V1 * cos(theta1)
= 1.35 m/s * cos(45 degrees)
= 1.35 m/s * 0.707 (rounded to three decimal places)
≈ 0.955 m/s
Vy1 = V1 * sin(theta1)
= 1.35 m/s * sin(45 degrees)
= 1.35 m/s * 0.707 (rounded to three decimal places)
≈ 0.955 m/s
The time taken by canoeist 1 to reach the island can be calculated by dividing the distance by its x-component velocity:
t1 = distance / Vx1
Given that the distance is 1.5 km and 1 km = 1000 m, we can convert it to meters:
distance = 1.5 km * 1000 m/km
= 1500 m
Now we have all the values to calculate the time taken by canoeist 1:
t1 = 1500 m / 0.955 m/s
≈ 1570.680 seconds
Now, to find the required speed for canoeist 2, we need to consider that canoeist 2 starts on the opposite shore of the lake, 1.5 km due east of canoeist 1. The distance from the opposite shore to the island will be the same as the distance from canoeist 1 to the island.
Therefore, the distance for canoeist 2 is also 1500 m.
To find the speed required for canoeist 2, we can divide the distance by the time calculated for canoeist 1:
V2 = distance / t1
V2 = 1500 m / 1570.680 s
= 0.955 m/s
So, canoeist 2 must have a speed of approximately 0.955 m/s to arrive at the island at the same time as canoeist 1.