Final answer:
The average horizontal speed of the grasshopper as it progresses down the road is calculated using the range formula for a projectile. After determining the grasshopper's initial velocity, the horizontal component of this velocity is found, and the average speed is obtained by dividing the range by the flight time. The average horizontal speed is approximately 0.72 m/s.
Step-by-step explanation:
To calculate the average horizontal speed of the grasshopper as it progresses down the road, we need to consider both the distance of each hop and the time it takes to complete one hop. Since the grasshopper achieves a range R of 1.0 m per hop, we can find the average speed by using the formula for the range of a projectile launched at an angle θ: R = (v^2 sin(2θ)) / g, where v is the initial velocity, θ is the launch angle, and g is the acceleration due to gravity (9.81 m/s^2).
We can rearrange the formula to solve for v, the initial velocity: v = sqrt(Rg / sin(2θ)). Plugging in the values, we get v = sqrt(1.0 m * 9.81 m/s^2 / sin(2 * 55 degrees)). The horizontal component of this velocity, which is what we are interested in, is v_x = v*cos(θ). Substituting the value of θ = 55 degrees, we obtain v_x = v*cos(55 degrees).
Finally, we find speed by dividing the range by the flight time of the hop. The time of flight, T, can be found from T = 2 * v * sin(θ) / g. The average horizontal speed is then R / T, which equals (1.0 m) / (2 * v * sin(55 degrees) / 9.81 m/s^2), resulting in an average horizontal speed of approximately 0.72 m/s, which is answer choice (b).