A small dog is trained to jump straight up a distance of 1.2 m. How much kinetic energy does the 7.2-kg dog need to jump this high? (The acceleration due to gravity is 9.8 m/s2.) Show your work.

To determine the kinetic energy required for the small dog to jump a distance of 1.2 m, we need to use the formula for gravitational potential energy.

Gravitational potential energy (PE) is given by the formula:
PE = mgh

where m is the mass of the object (dog), g is the acceleration due to gravity, and h is the height.

Given:
Mass of the dog (m) = 7.2 kg
Height (h) = 1.2 m
Acceleration due to gravity (g) = 9.8 m/s^2

Substituting the given values into the formula, we can calculate the potential energy (PE):
PE = (7.2 kg) * (9.8 m/s^2) * (1.2 m)

PE = 84.096 J (joules)

Therefore, the dog needs 84.096 joules of kinetic energy to jump a distance of 1.2 m.

To calculate the kinetic energy needed for the dog to jump, we need to use the formula for kinetic energy:

Kinetic Energy = 1/2 * mass * velocity^2

First, we need to find the velocity of the dog when it jumps. We can use the concept of potential energy to do that.

The potential energy of the dog at the highest point of its jump will be equal to the kinetic energy it had when it left the ground. This is because energy is conserved in a closed system.

Potential Energy = Kinetic Energy

At the top of the jump, the dog will have converted all of its initial kinetic energy into potential energy, given by the formula:

Potential Energy = mass * gravitational acceleration * height

Plugging in the values given:
Potential Energy = 7.2 kg * 9.8 m/s^2 * 1.2 m

Now that we know the potential energy, we can equate it to the kinetic energy of the dog:

Potential Energy = 1/2 * mass * velocity^2

Plugging in the values again:
7.2 kg * 9.8 m/s^2 * 1.2 m = 1/2 * 7.2 kg * velocity^2

Simplifying the equation:
84.96 J = 3.6 kg * velocity^2

Now, we can isolate the velocity:
velocity^2 = 84.96 J / 3.6 kg

Calculating:
velocity^2 ≈ 23.6 m^2/s^2

Finally, we take the square root of both sides to find the velocity:
velocity ≈ √23.6 m^2/s^2

Now that we have the velocity, we can calculate the kinetic energy using the initial formula:

Kinetic Energy = 1/2 * mass * velocity^2

Plugging in the values:
Kinetic Energy = 1/2 * 7.2 kg * (√23.6 m^2/s^2)^2

Simplifying:
Kinetic Energy = 1/2 * 7.2 kg * 23.6 m^2/s^2

Calculating:
Kinetic Energy ≈ 79.776 J

Therefore, the kinetic energy the dog needs to jump 1.2 m is approximately 79.776 joules.