A sled is being pulled across a horizontal patch of snow. Friction is negligible. The pulling force points in the same direction as the sled's displacement, which is along the +x axis. As a result, the kinetic energy of the sled increases by 47.5 percent. By what percentage would the sled's kinetic energy have increased if this force had pointed 66.5 ° above the +x axis?
To determine the percentage increase in the sled's kinetic energy when the pulling force points 66.5° above the +x axis, we need to understand the relationship between the angle and the change in kinetic energy.
The work done on an object (in this case, the sled) by the applied force is given by the formula:
Work = Force * Displacement * cos(θ)
Where:
- Work is equal to the change in kinetic energy (ΔKE)
- Force is the magnitude of the applied force
- Displacement is the magnitude of the sled's displacement
- θ is the angle between the applied force and the displacement vector
We can rewrite this equation to solve for the change in kinetic energy (ΔKE):
ΔKE = Work = Force * Displacement * cos(θ)
Now, we can calculate the change in kinetic energy when the pulling force points 66.5° above the +x axis. Let's assume the initial kinetic energy is KE.
ΔKE = KE * (47.5/100) [Given: Kinetic energy increased by 47.5%]
Let's solve for KE:
KE = ΔKE / (47.5/100)
Now, we can calculate the new change in kinetic energy when the pulling force points 66.5° above the +x axis. Let's call it ΔKE_new.
ΔKE_new = KE * (Force * Displacement * cos(θ)) / (Force * Displacement * cos(66.5°))
Since we have already calculated KE and ΔKE, we can substitute those values in the equation.
ΔKE_new = (ΔKE / (47.5/100)) * (cos(θ) / cos(66.5°))
Finally, we can calculate the percentage increase in kinetic energy:
Percentage Increase = (ΔKE_new - ΔKE) / ΔKE * 100
By substituting the values and calculating the expression, we can determine the percentage increase in the sled's kinetic energy when the force points 66.5° above the +x axis.