Why did the grasshopper jump 1.0 m in the first place? It must have been trying to avoid a lecture about the importance of staying on the ground! Talk about motivation!
But let's get serious for a moment and crunch the numbers. To find the speed of the grasshopper, we need to break down the initial velocity into its horizontal and vertical components.
The initial velocity can be split into two parts: the horizontal velocity (Vx) and the vertical velocity (Vy). We can use basic trigonometry to find these components.
Given that the grasshopper jumps at a 47° angle with the horizontal, we can find:
Vx = V * cos(47°)
Vy = V * sin(47°)
Now, since the grasshopper starts from rest, the horizontal component of its velocity (Vx) will remain constant throughout its flight. Therefore, Vx = 0 m/s.
So, we only need to focus on the vertical component of the velocity (Vy). From its initial rest position, it will travel upwards until it reaches its maximum height, at which point its vertical velocity will become zero. We can use kinematic equations to find this maximum height.
First, let's find the time it takes for the grasshopper to reach its maximum height (tmax). We can use the vertical component of the velocity (Vy) and the acceleration due to gravity (g = 9.8 m/s^2):
Vy = V * sin(47°)
tmax = Vy / g
Next, we can use this time to find the maximum height (hmax):
hmax = (Vy^2) / (2 * g)
And there you have it! The speed of the grasshopper is Vy = V * sin(47°), and the maximum height reached is hmax = (Vy^2) / (2 * g). Just be glad the grasshopper didn't choose a career in high jump – it might have given Usain Bolt a run for his money!