A skier is gliding along at 2.5m/s on horizontal, frictionless snow. He suddenly starts down a 10∘ incline. His speed at the bottom is 13m/s .

What is the length of the incline?

How long does it take him to reach the bottom?

Well, this skier seems to be in quite a downhill rush! Let's see if I can help calculate the length of the incline and the time it takes for him to reach the bottom.

To find the length of the incline, we can use the kinematic equation relating initial velocity, final velocity, acceleration, and displacement. In this case, the acceleration is due to gravity.

We have an initial velocity of 2.5 m/s and a final velocity of 13 m/s, and the acceleration due to gravity is approximately 9.8 m/s². We can use the equation:

v² = u² + 2as

where v is the final velocity, u is the initial velocity, a is the acceleration, and s is the displacement (length of the incline).

Plugging in the given values, we get:

13² = 2.5² + 2 * 9.8 * s

Solving for s, we find:

s ≈ 6.63 meters

So, the length of the incline is approximately 6.63 meters.

Now, let's move on to calculating the time it takes for the skier to reach the bottom. Since we have the initial velocity, final velocity, acceleration, and displacement, we can also use another kinematic equation:

v = u + at

where t represents time. Rearranging the equation, we get:

t = (v - u) / a

Plugging in the values, we have:

t = (13 - 2.5) / 9.8

Calculating this, we find:

t ≈ 1.19 seconds

So, it takes approximately 1.19 seconds for our skier to reach the bottom of the incline.

Hope that helps, and may your skiing adventures be as smooth as a buttered snowball!

To determine the length of the incline, we can start by analyzing the initial and final speeds of the skier.

Given:
Initial speed (v0) = 2.5 m/s
Final speed (v) = 13 m/s

We can use the principles of conservation of energy to find the length of the incline. On the horizontal surface, the skier has only kinetic energy (KE = (1/2)mv^2). As the skier moves down the incline, the gravitational potential energy (PE = mgh) is converted into kinetic energy.

Since the horizontal surface is frictionless, the total energy remains constant. Therefore, we can equate the initial kinetic energy with the final kinetic energy:

(1/2)m(v0)^2 = (1/2)mv^2

Simplifying the equation, we have:
(v0)^2 = v^2

Substituting the given values:
(2.5)^2 = 13^2

6.25 = 169

This equation is not true, which means we made a mistake in our calculations or assumptions. Let's reconsider the approach.

Since the skier is gliding on a frictionless, horizontal surface before going down the incline, the horizontal speed (2.5m/s) remains constant. Therefore, we do not need to consider it in our calculations.

Using the concept of conservation of energy, we can consider only the change in gravitational potential energy (mgh) and the change in kinetic energy (KE = (1/2)mv^2) as the skier moves down the incline.

Starting with the change in gravitational potential energy:
ΔPE = mgh

The length of the incline can be obtained using the trigonometry relationship:
h = l*sin(θ)

where h is the vertical height or the change in height of the incline, l is the length of the incline, and θ is the angle of the incline.

Substituting h = l*sin(θ) into the gravitational potential energy equation, we have:
ΔPE = mg*l*sin(θ)

Since the skier is gliding on a frictionless surface, the change in potential energy is equal to the change in kinetic energy:
KE = (1/2)mv^2

Now, we can equate the change in gravitational potential energy to the change in kinetic energy:
ΔPE = KE

mg*l*sin(θ) = (1/2)mv^2

Canceling out the common terms, we have:
gl*sin(θ) = (1/2)v^2

Substituting the given values:
9.8 * l * sin(10∘) = (1/2) * (13^2)

Simplifying the equation, we find:
l = (1/2) * (13^2) / (9.8 * sin(10∘))

l ≈ 15.7m

Therefore, the length of the incline is approximately 15.7 meters.

To determine the time it takes for the skier to reach the bottom of the incline, we can use the kinematic equation:

v = u + at

Where:
v = final velocity (13 m/s)
u = initial velocity (2.5 m/s)
a = acceleration (due to gravity, g, acting along the incline)
t = time

Since the skier is moving downward, we take the acceleration as:
a = g * sin(θ)

Let's calculate the acceleration:
a = 9.8 * sin(10∘)

Substituting the given values into the kinematic equation, we have:
13 = 2.5 + (9.8 * sin(10∘)) * t

Simplifying the equation, we find:
t = (13 - 2.5) / (9.8 * sin(10∘))

t ≈ 1.34s

Therefore, it takes approximately 1.34 seconds for the skier to reach the bottom of the incline.

5.0 s

acceleration= g*sin10

vf^2=vi^2+2d a
solve for d

average speed going down=(13+2.5 )/2

timedown=distance/avgspeed