really stuck, a 90 kg skier has 140 KJ of total mechanical energy when he is at a height of 120m above the bottom of the hill. calculate the skiers kinetic energy. calculate his speed?
Well, well, well, looks like the skier is up for a roller coaster ride! Let's crunch some numbers, shall we?
First, let's find the potential energy at the top of the hill. We'll multiply the mass (90 kg) by the gravitational acceleration (9.8 m/s²) and the height (120 m):
Potential energy = mass × gravitational acceleration × height
= 90 kg × 9.8 m/s² × 120 m
That gives us a whopping potential energy of... *drumroll*... 105,840 J!
But wait, there's more! We know that the total mechanical energy (which is the sum of potential and kinetic energy) is equal to 140 kJ (or 140,000 J). Since we just found the potential energy, we can now calculate the kinetic energy.
Kinetic energy = Total mechanical energy - Potential energy
= 140,000 J - 105,840 J
So, the skier's kinetic energy is 34,160 J! Phew, he's got quite the energy going on.
Now, to find the speed, we can use the equation for kinetic energy:
Kinetic energy = (1/2) × mass × velocity²
By rearranging that equation, we can solve for velocity:
Velocity = square root of ((2 × Kinetic energy) / mass)
= square root of ((2 × 34,160 J) / 90 kg)
After crunching those numbers, we get a speed of approximately... *drumroll*... 21 m/s!
So, hold on tight, Mr. Skier, because you'll be zooming down that hill at about 21 m/s! Enjoy the ride and stay funny!
To calculate the skier's kinetic energy and speed, we can use the principle of conservation of energy, which states that the total mechanical energy of an object is conserved (remains constant) as long as no external forces are acting on it.
The total mechanical energy (E) of the skier is the sum of his potential energy (PE) and kinetic energy (KE):
E = PE + KE
We are given that the skier has a total mechanical energy (E) of 140 kJ (kilojoules) when at a height (h) of 120 m above the bottom of the hill. We can calculate the potential energy using the formula:
PE = m * g * h
where m is the mass of the skier, g is the acceleration due to gravity, and h is the height.
Given:
m (mass of skier) = 90 kg
g (acceleration due to gravity) = 9.8 m/s^2
h (height) = 120 m
Now, let's calculate the potential energy (PE):
PE = m * g * h
= 90 kg * 9.8 m/s^2 * 120 m
= 105,840 J (joules)
= 105.84 kJ (kilojoules)
Using the conservation of energy principle, we can calculate the kinetic energy (KE):
KE = E - PE
= 140 kJ - 105.84 kJ
= 34.16 kJ (kilojoules)
Finally, to find the skier's speed (v), we can use the equation for kinetic energy:
KE = (1/2) * m * v^2
Given:
KE = 34.16 kJ (kilojoules)
m (mass of skier) = 90 kg
Now, let's solve for v (speed):
34.16 kJ = (1/2) * 90 kg * v^2
v^2 = (34.16 kJ * 2) / 90 kg
v^2 = 0.758 kJ/kg
v = square root of 0.758 kJ/kg
v ≈ 0.871 m/s
Therefore, the skier's kinetic energy is approximately 34.16 kJ, and his speed is approximately 0.871 m/s.
To calculate the skier's kinetic energy, we first need to find the potential energy at the given height, then subtract it from the total mechanical energy to get the kinetic energy.
Step 1: Calculate potential energy
The potential energy (PE) of an object at a certain height is given by the formula:
PE = mass x acceleration due to gravity x height
In this case, the mass of the skier is 90 kg, the acceleration due to gravity is 9.8 m/s², and the height is 120 m.
PE = 90 kg x 9.8 m/s² x 120 m
PE = 105,840 J
Step 2: Calculate kinetic energy
The kinetic energy (KE) is equal to the total mechanical energy (TME) minus the potential energy (PE):
KE = TME - PE
Given that the total mechanical energy (TME) is 140,000 J, we can substitute the values:
KE = 140,000 J - 105,840 J
KE = 34,160 J
So, the skier's kinetic energy is 34,160 J.
Step 3: Calculate speed
To find the speed of the skier, we can use the equation that relates kinetic energy and speed:
KE = 0.5 x mass x speed²
Substituting the values, we have:
34,160 J = 0.5 x 90 kg x speed²
Rearranging the equation to solve for speed:
speed² = (34,160 J) / (0.5 x 90 kg)
speed² = 759.11 m²/s²
Taking the square root of both sides, we find:
speed = √759.11 m²/s²
speed ≈ 27.6 m/s
Therefore, the skier's speed is approximately 27.6 m/s.