A student is skateboarding down a ramp that is 5.5 m long and inclined at 19° with respect to the horizontal. The initial speed of the skateboarder at the top of the ramp is 2.9 m/s. Neglect friction and find the speed at the bottom of the ramp.

Why did the skateboarder become an accountant? Because they wanted to stay balanced and keep their finances in check! Now, let's calculate the speed at the bottom of the ramp.

First, we need to find the vertical component of the initial velocity. We'll use the formula v₀y = v₀ * sin(angle).

v₀y = 2.9 m/s * sin(19°)
v₀y ≈ 0.988 m/s

Now, let's find the vertical displacement using the formula y = v₀y * t + 0.5 * a * t².
Since there's no vertical acceleration or time given, we'll need to find those first.

The vertical component of gravity is given by a = g * sin(angle), where g is the acceleration due to gravity (9.8 m/s²).
a = 9.8 m/s² * sin(19°)
a ≈ 3.28 m/s²

Next, we'll find the time it takes for the skateboarder to go down the ramp. We'll use the formula y = v₀y * t + 0.5 * a * t² and solve for t.

0.5 * a * t² = y
0.5 * 3.28 m/s² * t² = 5.5 m

Simplifying, we get:
1.64 m/s² * t² = 5.5 m
t² ≈ (5.5 m) / (1.64 m/s²)
t ≈ sqrt(3.35 s²)
t ≈ 1.83 s

Now that we have the time, let's find the horizontal component of the displacement using the formula x = v₀x * t.
Since there's no horizontal acceleration or initial horizontal velocity, we can assume v₀x = 0 m/s.

x = 0 m/s * 1.83 s
x = 0 m

Therefore, the horizontal displacement is 0 m.

Finally, we can find the speed at the bottom of the ramp using the Pythagorean theorem.

v = √(v₀x² + v₀y²)
v = √(0 m/s + (2.9 m/s)²)
v ≈ √(0 m/s² + 8.41 m²/s²)
v ≈ √8.41 m²/s²
v ≈ 2.9 m/s

Therefore, the speed at the bottom of the ramp is approximately 2.9 m/s. Keep rolling and stay balanced, just like a well-managed bank account!

To find the speed at the bottom of the ramp, we can use the concept of conservation of energy. The initial potential energy of the skateboarder at the top of the ramp will be converted into kinetic energy at the bottom.

First, let's find the height of the ramp. We can use the trigonometric relationship between the angle of inclination and the height:

height = length * sin(angle)
= 5.5 m * sin(19°)
≈ 1.84 m

Now, let's calculate the initial potential energy of the skateboarder:

potential energy = mass * gravity * height

Since the mass is not given, we can assume a standard value of 70 kg for the skateboarder:

potential energy = 70 kg * 9.8 m/s^2 * 1.84 m
≈ 1270 J

The conservation of energy states that the initial potential energy is equal to the final kinetic energy:

potential energy = kinetic energy

At the bottom of the ramp, the skateboarder only has kinetic energy:

kinetic energy = 1/2 * mass * velocity^2

Now, rearranging the equation and solving for the velocity:

velocity^2 = (2 * potential energy) / mass

We can substitute the known values:

velocity^2 = (2 * 1270 J) / 70 kg
velocity^2 ≈ 36.29 m^2/s^2

Taking the square root of both sides of the equation to solve for velocity:

velocity ≈ √36.29 m^2/s^2
velocity ≈ 6.02 m/s

Therefore, the speed at the bottom of the ramp is approximately 6.02 m/s.

thank you

where did you get 1.791 from

Apply conservation of energy. Neglect the rotational kinetic energy of the wheels, which will make a small difference since the wheels weigh much less than the skateboarder.

(M/2)[V2^2 - V1^2] = M g deltaH
= M g *5.5 sin 19

V1 = 2.9 m/s

Cancel the M's and solve for V2, the final velocity.

V2^2 - (2.9)^2 = 2*9.81*1.791 = 35.13
V2^2 = 35.13 + 8.41
V2 = 6.60 m/s