A block is given an initial velocity of 4.00 m/s up a frictionless incline of angle θ = 21.0°. How far up the incline does the block slide before coming to rest?
Vo = 4m/s[21o].
Yo = 4*sin21 = 1.43 m/s. = Ver. component of initial velocity
Y^2 = Yo^2 + 2g*h = 0 @ max htg.
(1.43)^2 - 19.6h = 0
h = 0.104 m.
d = h/sin21 = 0.104/sin21 = 0.291 m. up incline.
How’d u get g, for 19.6?
To determine how far up the incline the block slides before coming to rest, we can use the principles of physics, such as the components of motion and the conservation of energy.
Step 1: Analyze the Given Information
- Initial velocity (vi) of the block = 4.00 m/s
- Incline angle (θ) = 21.0°
- The incline is frictionless
Step 2: Resolve the Forces into Components
The weight (W) of the block acts vertically downward and can be resolved into two components:
- Perpendicular to the incline (W⊥) = W * cos(θ)
- Parallel to the incline (W∥) = W * sin(θ)
Step 3: Calculate the Acceleration of the Block
Since the incline is frictionless, the net force acting on the block is the component of gravity parallel to the incline:
F∥ = m * a,
where m is the mass of the block and a is the acceleration of the block.
Using Newton's second law (F = ma) and substituting the parallel component of weight into the equation, we have:
m * a = m * g * sin(θ)
a = g * sin(θ)
where g is the acceleration due to gravity (approximately 9.8 m/s²).
Step 4: Calculate the Distance Traveled
The distance traveled by the block up the incline can be determined using the equations of motion. In this case, we'll use the equation:
v² = vi² + 2 * a * d,
where v is the final velocity (which is zero since the block comes to rest), vi is the initial velocity (4.00 m/s), a is the acceleration (-g * sin(θ)), and d is the distance traveled.
Rearranging the equation to solve for d:
d = (v² - vi²) / (2 * a)
Since v = 0, the equation becomes:
d = -vi² / (2 * a)
Substituting the values, we have:
d = -(4.00 m/s)² / (2 * (-g * sin(θ)))
Step 5: Calculate the Result
Plug in the values and calculate:
d = - (4.00 m/s)² / (2 * (9.8 m/s²) * sin(21.0°))
Using a calculator, the result is:
d ≈ -1.34 m
However, distance cannot be negative. Therefore, the absolute value of the result gives the distance traveled by the block up the incline before coming to rest:
d ≈ 1.34 m (rounded to two decimal places)
Therefore, the block slides approximately 1.34 meters up the incline before coming to rest.
To find how far up the incline the block slides before coming to rest, we need to use the concepts of energy conservation.
Since the incline is frictionless, the only force acting on the block is its weight, which is acting down the incline. The gravitational potential energy (GPE) will be converted into kinetic energy (KE) as the block slides up the incline, and then the kinetic energy will be converted back into gravitational potential energy as the block slows down and comes to rest.
We can calculate the distance the block slides using the following steps:
Step 1: Determine the initial kinetic energy (KE_initial) of the block.
The formula for kinetic energy is:
KE = (1/2) * m * v^2,
where
m = mass of the block,
v = initial velocity of the block.
In this case, the initial velocity of the block is given as 4.00 m/s.
Step 2: Calculate the height (h) to which the block rises.
To do this, we need to consider changes in potential energy.
ΔPE = m * g * h,
where
m = mass of the block,
g = acceleration due to gravity,
h = height (vertical distance) the block rises.
Since the block comes to rest at its highest point, the final kinetic energy (KE_final) will be zero. Therefore, the change in kinetic energy (∆KE) is given by:
∆KE = KE_final - KE_initial = 0 - KE_initial = -KE_initial
Step 3: Equate the change in kinetic energy (∆KE) with the change in potential energy (∆PE) to find the height (h) to which the block rises.
∆KE = ∆PE,
-KE_initial = m * g * h
Step 4: Solve for the height (h):
h = -KE_initial / (m * g)
Given that the inclination angle is θ = 21.0°, we can relate the vertical height (h) to the distance up the incline (d) using the trigonometric relation:
h = d * sin(θ),
where
d = distance traveled up the incline,
θ = angle of the incline.
Step 5: Substitute the value of h into the equation and solve for d.
I hope this explanation helps you understand how to approach and solve the problem!