What is the average horizontal speed of a grasshopper that hops along a road, launching itself at an angle of 45 degrees and achieving a range of 0.78 meters?
The average horizontal speed of the grasshopper is 0.707.
Calculating the Average Horizontal Speed
The average horizontal speed of the grasshopper can be determined by first calculating its initial velocity in the horizontal direction. With the launch angle being 45 degrees, we can find the initial velocity using the formula V₀ = R / (cos(θ₀)), where R is the range achieved by the grasshopper. Substituting the given values, we get V₀ = 0.78 / (cos(45⁰)).
Next, we calculate the average speed using the formula average speed = total distance / total time. In this case, the total distance is equal to the range R, and the total time can be found by dividing the range by the horizontal velocity V₀. Therefore, the average horizontal speed is equal to R / (R / (cos(θ₀))).
Simplifying the expression further, we get the average horizontal speed = cos(45⁰). Since the launch angle is 45 degrees, the average horizontal speed of the grasshopper is cos(45⁰) = 0.707 (rounded to three decimal places).
This value signifies the average horizontal speed of the grasshopper as it hops along the road at a launch angle of 45 degrees and achieves a range of 0.78 meters.