🤓 Based on our data, we think this question is relevant for Professor Lai's class at TEXAS.
For this problem, we're looking for the initial speed and the range of the froghopper's leap given the direction of its launch and the maximum height it has to travel.
Since the takeoff and landing are at the same height, this is a symmetrical launch problem.
For projectile motion problems in general, we'll follow these steps to solve:
The four UAM (kinematics) equations are:
In our coordinate system, the +y-axis is pointing upwards and the +x-direction is horizontal along the launch direction. That means ay = −g, and ax = 0 (because the only acceleration acting on a projectile once it's launched is gravity.)
For the special case of a symmetrical launch, we also have equations for the horizontal distance traveled, also called the range, and maximum height of the projectile:
The froghopper, Philaenus spumarius, holds the world record for insect jumps. When leaping at an angle of 58.0° above the horizontal, some of the tiny critters have reached a maximum height of 58.7 cm above the level ground.
(a) What was the takeoff speed for such a leap?
(b) What horizontal distance did the froghopper cover for this world-record leap?