Question

Two identical grasshoppers jump into the air with the same initial speed and experience no air resistance. Grasshopper A goes straight up, but grasshopper B goes up at a 66° angle above the horizontal. Which of the following statements about these grasshoppers are correct? (There could be more than one correct choice.
Choices below:

a.) At their highest point, both of them have the same amount of mechanical energy.
b.) At their highest point, grasshopper B is moving faster than grasshopper A.
c.) At their highest point, both of them have the same amount of kinetic energy.
d.) At their highest point, both of them have the same amount of gravitational potential energy.
e.)At their highest point, grasshopper A has more gravitational potential energy than grasshopper B.

Answer 1

Answer:

a.) At their highest point, both of them have the same amount of mechanical energy.

Explanation:

Grasshopper A jumps upwards with speed v while grasshopper B jumps at an angle of 66 degree with horizontal at same speed

so vertical speed of B is given

$v_y = v sin66 = 0.91v$

$v_x = v cos66 = 0.41v$

So when both the grasshopper will jump upwards they both will attain different maximum heights

Since initial speed of both grasshopper is same so their mechanical energy is always conserved

So mechanical energy of grasshopper A is given as

$U = \frac{1}{2}mv^2$

Mechanical energy of grasshopper B is given as

$U = \frac{1}{2}m(vcos^2\theta) + \frac{1}{2}m(vsin\theta)^2$

$U = \frac{1}{2}mv^2$

So we have

a.) At their highest point, both of them have the same amount of mechanical energy.