Two frisky grasshoppers collide in midair at the top of their respective trajectories and grab onto each other, holding tight thereafter. One is a robust 250 g beast initially moving south at 20.0 cm/s, while the other is a svelte 150 g creature initially moving north at 60.0 cm/s. Calculate the decrease in kinetic energy that results from the collision.
Find the final combined velocity after the collision using conservation of momentum,
m1v1+m2v2 = (m1+m2)v
solve for v.
Compare kinetic energy before and after impact,
Before: (1/2)m1v1²+(1/2)m2v2²
after: (1/2)(m1+m2)v²
1008176.54J
To calculate the decrease in kinetic energy, we need to find the initial kinetic energy of both grasshoppers before the collision and compare it to the final kinetic energy after the collision.
The formula to calculate kinetic energy is:
Kinetic Energy = (1/2) * mass * velocity^2
For the first grasshopper with a mass of 250 g and an initial velocity of 20.0 cm/s:
Initial Kinetic Energy (Grasshopper 1) = (1/2) * 0.25 kg * (0.2 m/s)^2
Converting the mass to kilograms:
Initial Kinetic Energy (Grasshopper 1) = (1/2) * 0.25 kg * (0.2 m/s)^2
Initial Kinetic Energy (Grasshopper 1) = 0.025 J
For the second grasshopper with a mass of 150 g and an initial velocity of 60.0 cm/s:
Initial Kinetic Energy (Grasshopper 2) = (1/2) * 0.15 kg * (0.6 m/s)^2
Converting the mass to kilograms:
Initial Kinetic Energy (Grasshopper 2) = (1/2) * 0.15 kg * (0.6 m/s)^2
Initial Kinetic Energy (Grasshopper 2) = 0.027 J
Total initial kinetic energy before the collision:
Total Initial Kinetic Energy = Initial Kinetic Energy (Grasshopper 1) + Initial Kinetic Energy (Grasshopper 2)
Total Initial Kinetic Energy = 0.025 J + 0.027 J
Total Initial Kinetic Energy = 0.052 J
After the grasshoppers collide and hold onto each other, they have a final velocity of 0 m/s since they remain stationary.
Final Kinetic Energy = (1/2) * (0.25 kg + 0.15 kg) * (0 m/s)^2
Final Kinetic Energy = 0 J
The decrease in kinetic energy is then:
Decrease in Kinetic Energy = Total Initial Kinetic Energy - Final Kinetic Energy
Decrease in Kinetic Energy = 0.052 J - 0 J
Decrease in Kinetic Energy = 0.052 J
Therefore, the decrease in kinetic energy resulting from the collision is 0.052 J.
To calculate the decrease in kinetic energy resulting from the collision, we need to find the initial kinetic energy of both grasshoppers and the final kinetic energy after the collision.
The formula for kinetic energy is:
KE = 0.5 * m * v^2
Where:
KE = Kinetic Energy
m = Mass
v = Velocity
Let's start by finding the initial kinetic energy of both grasshoppers:
For the robust grasshopper:
Mass (m1) = 250 g = 0.25 kg
Velocity (v1) = -20.0 cm/s (negative sign indicates it is moving south)
KE1 = 0.5 * m1 * v1^2
= 0.5 * 0.25 kg * (-20.0 cm/s)^2
Since the velocity is squared, we don't need to consider the negative sign. So,
KE1 = 0.5 * 0.25 kg * (20.0 cm/s)^2
For the svelte grasshopper:
Mass (m2) = 150 g = 0.15 kg
Velocity (v2) = 60.0 cm/s (positive sign indicates it is moving north)
KE2 = 0.5 * m2 * v2^2
= 0.5 * 0.15 kg * (60.0 cm/s)^2
Now, let's find the final kinetic energy after the collision. Since the grasshoppers grab onto each other and hold tight, they move together as one body after the collision. We need to find the final velocity (vf) of the combined system:
Total mass (mtotal) = m1 + m2 = 0.25 kg + 0.15 kg = 0.40 kg
To find the final velocity, we can use the conservation of momentum principle, which states that the total momentum before and after the collision is equal:
m1 * v1 + m2 * v2 = mtotal * vf
(0.25 kg * -20.0 cm/s) + (0.15 kg * 60.0 cm/s) = 0.40 kg * vf
Simplify the equation:
-5.0 kg·cm/s + 9.0 kg·cm/s = 0.40 kg · vf
4.0 kg·cm/s = 0.40 kg · vf
Divide both sides by 0.40 kg:
vf = 4.0 kg·cm/s / 0.40 kg
vf = 10.0 cm/s (negative sign indicates it is moving south)
Now, let's calculate the final kinetic energy (KEf) using the combined mass (mtotal) and final velocity (vf):
KEf = 0.5 * mtotal * vf^2
= 0.5 * 0.40 kg * (-10.0 cm/s)^2
Again, we don't need to consider the negative sign since velocity is squared. So,
KEf = 0.5 * 0.40 kg * (10.0 cm/s)^2
Now, we can calculate the decrease in kinetic energy (ΔKE) in the collision:
ΔKE = KEi - KEf
Substituting the values we calculated earlier:
ΔKE = [0.5 * 0.25 kg * (20.0 cm/s)^2 + 0.5 * 0.15 kg * (60.0 cm/s)^2] - [0.5 * 0.40 kg * (10.0 cm/s)^2]
After performing the calculations, you'll find the decrease in kinetic energy resulting from the collision.