Using the kinematic equations for free fall, we calculate the initial velocity of a hopper reaching a height of 1.3 meters. The answer closest to the calculated value, using approximate gravity, is option B. 4.9 m/s, which is consistent with the options provided.
The question is asking for the initial velocity of a hopper (or any object) that jumps straight up to a height of 1.3 meters and then falls back down, neglecting air resistance. To solve this, we can use the kinematic equation for free fall under gravity:
final velocity (v)² = initial velocity (u)² + 2 * acceleration (g) * height (h)
Since the final velocity is just before hitting the ground, it would be equal to the magnitude of the initial velocity but in the opposite direction. We take downward as positive, so our equation simplifies to:
0 = u² + 2 * g * h
We can reverse this to solve for the initial velocity as follows:
u² = -2 * g * h
Now we can plug in the acceleration due to gravity, g = 9.8 m/s², and the height, h = 1.3 m, to find u:
u² = -2 * (9.8 m/s²) * (1.3 m)
u = √(2 * 9.8 m/s² * 1.3 m)
u = √(25.48 m²/s²)
u = 5.047 m/s (approximately)
However, the options given don't include 5.047 m/s. Considering the closest answer to the calculated velocity and the fact that the question seems to be using the approximation g = 10 m/s², the initial velocity is most likely option B. 4.9 m/s, as it is closest to the calculated value when using the approximate value of gravity.
So, the initial velocity of the hopper just before reaching the ground is closest to option B. 4.9 m/s.