A 66 kg human sprinter can accelerate from rest to 11 m/s in 3.4s. During the same interval, a 35 kg greyhound can accelerate from rest to 20 m/s. Compute
(a) the change in kinetic energy of the sprinter.
(b) the change in kinetic energy of the greyhound.
(c) the average power output of the sprinter.
(d) the average power output of the greyhound.
Change in Kinetic Energy ΔK = Kf - Ki
Kf = 1/2 * m * vf^2
Ki = 1/2 * m * vi^2 (vi is zero, so Ki is zero)
Therefore ΔK = Kf - 0 = 1/2 * m * vf^2
Average power over a time interval Δt is ΔE/Δt .
ΔE = ΔK
Therefore Average Power = ΔK/Δt
Well, well, it seems like we have some fast and furious athletes here. Let's unravel the mysteries of their energy and power!
(a) To calculate the change in kinetic energy of the sprinter, we use the equation ΔKE = 1/2 * m * (vf^2 - vi^2), where ΔKE is the change in kinetic energy, m is the mass, vf is the final velocity, and vi is the initial velocity (which is zero since they start from rest). Plugging in the values, we get:
ΔKE = 1/2 * 66 kg * (11 m/s)^2 - 1/2 * 66 kg * (0 m/s)^2
ΔKE = 1/2 * 66 kg * 121 m^2/s^2 - 0
ΔKE = 3,663 J
So, the change in kinetic energy of the sprinter is 3,663 Joules.
(b) Now, let's calculate the change in kinetic energy of the greyhound. Using the same formula, we have:
ΔKE = 1/2 * 35 kg * (20 m/s)^2 - 1/2 * 35 kg * (0 m/s)^2
ΔKE = 1/2 * 35 kg * 400 m^2/s^2 - 0
ΔKE = 7,000 J
Hence, the change in kinetic energy of the greyhound is 7,000 Joules.
(c) To find the average power output of the sprinter, we can use the formula P = ΔKE / t, where P is the average power, ΔKE is the change in kinetic energy, and t is the time taken. Plugging in the values, we get:
P = 3,663 J / 3.4 s
P ≈ 1,078 Watts
So, the average power output of the sprinter is approximately 1,078 Watts.
(d) Finally, let's calculate the average power output of the greyhound:
P = 7,000 J / 3.4 s
P ≈ 2,059 Watts
The average power output of the greyhound is approximately 2,059 Watts.
And there you have it, my friend! The sprinter and the greyhound have some impressive energy changes and power outputs. Keep running and keep laughing!
To compute the change in kinetic energy, we can use the formula:
ΔKE = 1/2 * m * (vf^2 - vi^2)
where ΔKE is the change in kinetic energy, m is the mass of the object, vf is the final velocity, and vi is the initial velocity.
(a) For the sprinter:
m = 66 kg
vf = 11 m/s
vi = 0 m/s
ΔKE = 1/2 * 66 kg * (11 m/s)^2 - 0 = 1/2 * 66 kg * 121 m^2/s^2 = 3,786 J
The change in kinetic energy of the sprinter is 3,786 Joules.
(b) For the greyhound:
m = 35 kg
vf = 20 m/s
vi = 0 m/s
ΔKE = 1/2 * 35 kg * (20 m/s)^2 - 0 = 1/2 * 35 kg * 400 m^2/s^2 = 7,000 J
The change in kinetic energy of the greyhound is 7,000 Joules.
To compute the average power output, we can use the formula:
Power = ΔKE / t
where Power is the average power output, ΔKE is the change in kinetic energy, and t is the time interval.
(c) For the sprinter:
ΔKE = 3,786 J (from part a)
t = 3.4 s
Power = 3,786 J / 3.4 s = 1,114.11 W
The average power output of the sprinter is approximately 1,114.11 Watts.
(d) For the greyhound:
ΔKE = 7,000 J (from part b)
t = 3.4 s
Power = 7,000 J / 3.4 s = 2,058.82 W
The average power output of the greyhound is approximately 2,058.82 Watts.
To compute the change in kinetic energy and average power output for both the sprinter and the greyhound, we need to use the formulas for kinetic energy and power.
The formula for kinetic energy is:
K.E. = 0.5 * mass * velocity^2
The formula for average power is:
Power = Work / time
where Work is equal to the change in kinetic energy.
Let's calculate each part one by one:
(a) The change in kinetic energy of the sprinter can be calculated as follows:
Initial velocity (v1) = 0 m/s (as the sprinter starts from rest)
Final velocity (v2) = 11 m/s
Mass of the sprinter (m1) = 66 kg
Using the formula for kinetic energy:
K.E. = 0.5 * mass * velocity^2
Initial kinetic energy (K.E.1) = 0.5 * m * v1^2
Final kinetic energy (K.E.2) = 0.5 * m * v2^2
Change in kinetic energy = K.E.2 - K.E.1
Let's plug in the values:
Change in kinetic energy = 0.5 * 66 kg * (11 m/s)^2 - 0.5 * 66 kg * (0 m/s)^2
Simplifying the equation:
Change in kinetic energy = 0.5 * 66 kg * 121 m^2/s^2
(b) The change in kinetic energy of the greyhound can be calculated in the same way as the sprinter. Let's plug in the values:
Mass of the greyhound (m2) = 35 kg
Initial velocity (v1) = 0 m/s (as the greyhound starts from rest)
Final velocity (v2) = 20 m/s
Change in kinetic energy = 0.5 * 35 kg * (20 m/s)^2 - 0.5 * 35 kg * (0 m/s)^2
Simplifying the equation:
Change in kinetic energy = 0.5 * 35 kg * 400 m^2/s^2
(c) To calculate the average power output of the sprinter, we need the time it takes for the acceleration. The time given is 3.4 s. Using the formula for average power:
Power = Work / time
Substituting the values:
Power = (Change in kinetic energy of the sprinter) / 3.4 s
Substitute the value of the change in kinetic energy calculated in part (a) and divide it by 3.4 s to get the average power output of the sprinter.
(d) The calculation for the average power output of the greyhound is the same as for the sprinter. The change in kinetic energy is plugged into the equation with the given time interval of 3.4 s. Divide the change in kinetic energy calculated in part (b) by 3.4 s to get the average power output of the greyhound.
I hope this explanation helps you understand how to solve the problem!
Human Sprinter:
a = (V-Vo)/t = (11-0)/3.4 = 3.24 m/s^2.
d = 0.5a*t^2 = 1.62*(3.40)^2 = 16.96 m.
Fs = m*g = 66kg * 9.8N/kg = 646.8 N. =
Force of sprinter.
Greyhound Sprinter:
a = (20-0)/3.4 = 5.88 m/s^2.
d = 2.94*(3.4)^2 = 34 m.
Fg = 35kg * 9.8N/kg = 343 N. = Force of
Greyhound.
a. KE Change=0.5m*V^2-0.=33*11^2=3993 Joules.
b. KE Change=17.5*20^2=70,000 Joules.
c. Po = Fs*d/t.
Po = 646.8 * 16.96/3.4 = 3226 J/s.
d. Po = Fg*d/t.
Po = 343 * 34/3.4 = 3430 J/s.