Question

A grasshopper can jump a distance of 30 in (0.76 m) from a standing start.

a.) Assuming no air resistance,if the grasshopper takes off at the optimal angle 45 degree for maximum distance of the jump, what is the initial speed of the jump.

b.) Most animals jump at a lower angle than 45 degree. Suppose the grasshopper takes off at 30 degree from the horizontal. What jump speed is necessary to reach the noted distance?

Answer 1

Answer:

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Explanation:

The curve of the jumping grasshopper is similar to the trajectory of projectile starting with velocity v₀ and an angle θ to the horizontal:

y = x · tan(θ) - g·x² / (2·v₀² · cos²(θ))

The distance traveled until the grasshopper reaches the ground again (that means y=0) is:

d = 2·v₀·sin(2θ) / g

The maximum distance is jumped at an angle of θ=45°

d_max = 2·v₀ / g

<=>

v₀ = d_max·g/2

The horizontal velocity at which the grasshopper travels is

v = v₀ · cos(45°) = v₀/√2 = d_max·g/√8

If the grasshopper moves forward constantly at this speed he travels

s = v ·t =  d_max·g·t/√8

= 0.65m · 9.81m/s² · 3600s / √8

= 8116m