A small dog is trained to jump straight up a distance of 1.1 m. How much kinetic energy does the 7.7-kg dog need to jump this high? (The acceleration due to gravity is 9.8 m/s^2.) Show your work.
K.E. becomes P.E.
K.E. = m g h = 7.7 * 9.8 * 1.1 ... Joules
Thank you!
Well, if a small dog wants to jump that high, it better not have a fear of flying! Let's calculate the kinetic energy required.
The formula for kinetic energy is Ek = 0.5 * m * v^2, where Ek is the kinetic energy, m is the mass, and v is the velocity of the object.
First, we need to find the velocity. In this case, we're dealing with vertical motion, so we'll use the formula v = √(2 * g * h), where g is the acceleration due to gravity and h is the height.
Plugging in the values, v = √(2 * 9.8 m/s^2 * 1.1 m). That gives us v = √(21.56 m^2/s^2), which simplifies to v = 4.64 m/s.
Now we can plug the velocity into the kinetic energy formula: Ek = 0.5 * 7.7 kg * (4.64 m/s)^2.
Calculating that out, we get Ek ≈ 82.81 joules.
So, the hilarious answer is that the dog needs about 82.81 joules of energy, but maybe a good belly rub or a treat will also do the trick!
To calculate the kinetic energy required for the dog to jump this high, we need to use the conservation of energy principle. Here are the steps:
1. Determine the potential energy at the maximum height:
- The potential energy (PE) at the maximum height is given by the equation: PE = mgh, where m is the mass, g is the acceleration due to gravity, and h is the height.
- Substituting the given values: m = 7.7 kg and h = 1.1 m, and g = 9.8 m/s^2, we have: PE = (7.7 kg)(9.8 m/s^2)(1.1 m).
2. Calculate the kinetic energy at the maximum height:
- The kinetic energy (KE) at the maximum height is given by the equation: KE = PE.
- Substituting the calculated value from step 1, the kinetic energy is: KE = (7.7 kg)(9.8 m/s^2)(1.1 m).
3. Perform the calculation:
- Multiplying the values: KE = 7.7 kg * 9.8 m/s^2 * 1.1 m.
- The units cancel out, leaving us with the final answer: KE = 84.674 J.
Therefore, the dog needs approximately 84.674 Joules of kinetic energy to jump a distance of 1.1 m.
To find the kinetic energy required for the dog to jump, we can use the formula:
Kinetic Energy (KE) = 0.5 * mass * velocity^2
Since the dog is jumping straight up, its initial velocity is zero (as it starts from rest). Therefore, the velocity in the formula will be the final velocity, which can be determined using the equation of motion:
vf^2 = vi^2 + 2 * a * d
where
vf = final velocity (which is zero in this case),
vi = initial velocity (zero),
a = acceleration due to gravity (9.8 m/s^2), and
d = distance (1.1 m).
Rearranging the equation, we get:
0 = 0 + 2 * 9.8 * 1.1
Now, let's calculate the final velocity:
vf^2 = 2 * 9.8 * 1.1
vf^2 = 21.56
Taking the square root of both sides, we find:
vf ≈ 4.64 m/s
Now, we can substitute the values into the kinetic energy formula:
KE = 0.5 * mass * velocity^2
KE = 0.5 * 7.7 * (4.64)^2
Now, let's perform the calculation:
KE ≈ 0.5 * 7.7 * 21.5296
KE ≈ 82.5288 Joules
Therefore, the kinetic energy required for the 7.7-kg dog to jump 1.1 m is approximately 82.5288 Joules.