A 69.4-kg skier coasts up a snow-covered hill that makes an angle of 24.3° with the horizontal. The initial speed of the skier is 7.04 m/s. After coasting 1.78 m up the slope, the skier has a speed of 3.04 m/s. Calculate the work done by the kinetic frictional force that acts on the skis.
I've done this question 3 different times, CAPA doesn't accept my answer.
I've used Energy lost as friction = KElost - PEgained, getting 900 J. But this isn't right. I don't know what else to do.
PE gained= mg(sin24.3)d
friction work= unknown
Initial KE-finalPE=workdoneby friction+finalKE
1/2 m*7.04^2-m*g*1.78*sin24.3-1/2 m 3.04^2= work done be friction
1/2 *69.4*(7.04^2-3.04^2)-mg*1.78*sin24.3=workdone by friction.
To solve this problem, we need to calculate the work done by the kinetic frictional force. We can use the work-energy theorem to find the work done by the force of friction, which can be defined as:
Work done = Change in kinetic energy
The initial kinetic energy (KE_i) can be calculated using the mass (m) and initial speed (v_i) of the skier:
KE_i = 1/2 * m * v_i^2
The final kinetic energy (KE_f) can be calculated using the final speed (v_f):
KE_f = 1/2 * m * v_f^2
The work done by the frictional force is given by the difference in kinetic energy:
Work done = KE_f - KE_i
Let's substitute the given values into the equations:
m = 69.4 kg
v_i = 7.04 m/s
v_f = 3.04 m/s
KE_i = 1/2 * 69.4 kg * (7.04 m/s)^2
KE_i = 1723.67 J
KE_f = 1/2 * 69.4 kg * (3.04 m/s)^2
KE_f = 667.68 J
Work done = KE_f - KE_i
Work done = 667.68 J - 1723.67 J
Work done ≈ -1055 J
(Note: The negative sign indicates that work is done against the direction of the frictional force)
The work done by the kinetic frictional force is approximately -1055 J.
To solve this problem, let's start by breaking down the steps involved and reviewing the relevant concepts.
First, we need to calculate the change in potential energy as the skier coasts up the hill. We can use the formula:
ΔPE = m * g * Δh
where ΔPE is the change in potential energy, m is the mass of the skier (69.4 kg), g is the acceleration due to gravity (9.8 m/s²), and Δh is the change in height.
Since the skier coasts up the hill, the change in height is given by:
Δh = d * sin(θ)
where d is the distance traveled up the slope (1.78 m) and θ is the angle of the hill (24.3°).
Using this information, we can now calculate the change in potential energy:
ΔPE = (69.4 kg) * (9.8 m/s²) * (1.78 m) * sin(24.3°)
Next, we need to determine the change in kinetic energy of the skier. The initial kinetic energy (KE₁) is given by:
KE₁ = (1/2) * m * v₁²
where v₁ is the initial speed of the skier (7.04 m/s).
The final kinetic energy (KE₂) is given by:
KE₂ = (1/2) * m * v₂²
where v₂ is the final speed of the skier (3.04 m/s).
The change in kinetic energy is then:
ΔKE = KE₂ - KE₁
Substituting the given values into the equations, we can calculate ΔPE and ΔKE.
Lastly, we need to determine the work done by the kinetic frictional force. The work done by friction is equal to the change in mechanical energy of the skier. This can be written as:
Work = ΔKE + ΔPE
Substituting the calculated values of ΔKE and ΔPE, we can find the work done by the kinetic frictional force.
Make sure to double-check your calculations and units throughout the problem to ensure accuracy.