You pull a sled across a horizontal frictionless patch of snow. If your pulling force is in the same direction as the sled's displacement and increases the kinetic energy of the sled by 34%, by what percentage would the sled's kinetic energy increase if the pulling force acted at an angle of 42° above the horizontal?

W1 = F s = ∆KE = 0.34KE0

W2 = (F cos 42◦)s = F s cos 42◦ = 0.34∆KE0 cos 42◦ = 0.25∆KE0

Therefore, there is an 25% increase in kinetic energy.

Ah, so we're talking about some sled physics, huh? Well, let's slide into this question! When the pulling force is in the same direction as the sled's displacement, we get a 34% increase in kinetic energy. But what happens when the force acts at an angle of 42° above the horizontal? Well, hold on tight, because we're about to go on a wild slope!

When the force is at an angle, only the component of the force in the direction of the sled's displacement actually does any work. The perpendicular component just likes to chill out and do nothing. So, we need to find that parallel component. But don't worry, I won't make you do any trigonometry gymnastics! We can use the "cosine" to find it.

The cosine of the angle is the adjacent side divided by the hypotenuse. In this case, the adjacent side is the component of the force in the direction of the sled's displacement, and the hypotenuse is the total force. So, if we multiply the total force by the cosine of the angle, we'll get the parallel component.

Once we have that component, we can just apply the same logic as before. So, if the force at an angle of 42° above the horizontal increases the sled's kinetic energy by 34%... Are you ready for this punchline? Drumroll, please!

The sled's kinetic energy would increase by... approximately 23.32%! But hey, don't worry too much about the numbers. Just focus on enjoying the ride and spreading some laughter along the way!

To find the percentage increase in the sled's kinetic energy when the pulling force acts at an angle of 42° above the horizontal, we can use the principles of vector addition.

First, let's consider the case where the pulling force is in the same direction as the sled's displacement:

1. In this scenario, the full pulling force is responsible for the increase in kinetic energy, resulting in a 34% increase.

Now, let's analyze the case where the pulling force acts at an angle of 42° above the horizontal:

2. We can resolve the pulling force into two components: one parallel to the sled's displacement and one perpendicular to the sled's displacement.
- The component parallel to the sled's displacement will contribute to the sled's kinetic energy.
- The component perpendicular to the sled's displacement will not contribute to the sled's kinetic energy because it is acting perpendicular to the direction of motion.

To find the component of the pulling force parallel to the sled's displacement, we can calculate the cosine of the angle:

cos(42°) ≈ 0.7431

This means that approximately 74.31% of the pulling force is parallel to the sled's displacement.

Since the parallel component of the force is responsible for changing the sled's kinetic energy, the increase in kinetic energy can be calculated as follows:

Increase in kinetic energy = 0.7431 * 34%
Increase in kinetic energy ≈ 25.26%

Therefore, if the pulling force acts at an angle of 42° above the horizontal, the sled's kinetic energy would increase by approximately 25.26%.

To find the percentage by which the sled's kinetic energy would increase when the pulling force acts at an angle of 42° above the horizontal, we can use the concept of work and energy.

When the pulling force is in the same direction as the sled's displacement, all of the force contributes to the sled's kinetic energy. However, when the pulling force acts at an angle above the horizontal, only a component of the force contributes to the sled's displacement and thereby increases its kinetic energy.

To find this component of the force, we need to calculate the dot product between the force vector and the displacement vector. The dot product is given by the equation:

W = |F| |d| cosθ

where W is the work done, |F| is the magnitude of the force, |d| is the magnitude of the displacement, and θ is the angle between the force and displacement vectors.

In this case, let's assume the initial kinetic energy of the sled is K1, and when the pulling force is in the same direction as the sled's displacement, the kinetic energy increases by 34%, resulting in a final kinetic energy of K2 = K1 + 0.34K1 = 1.34K1.

Now, when the pulling force acts at an angle of 42° above the horizontal, only a component of the force contributes to the displacement. This component can be calculated using the cosine of the angle, so the effective force applied is |F| cosθ.

Using the work-energy theorem, we can equate the work done by this effective force to the change in kinetic energy:

W = ΔK
|F| |d| cosθ = ΔK
|F| cosθ = ΔK / |d|

Now, let's substitute the values we know. We have ΔK = 1.34K1 - K1 = 0.34K1, and we know the angle is 42°.

|F| cosθ = (0.34K1) / |d|

To determine the percentage increase in the sled's kinetic energy, we need to compare the change in kinetic energy ΔK to the initial kinetic energy K1. The percentage increase is given by:

Percentage increase = (ΔK / K1) * 100

Now, let's substitute the values we know into this equation:

Percentage increase = ((0.34K1) / K1) * 100

Simplifying further, we find:

Percentage increase = 34%

Therefore, when the pulling force acts at an angle of 42° above the horizontal, the sled's kinetic energy would increase by the same percentage as when the pulling force is in the same direction as the sled's displacement, which is 34%.