A block is given an initial velocity of 2.00 m/s up a frictionless 23.0° incline. How far up the incline does the block slide before coming (momentarily) to rest?
To find the distance the block slides up the incline before coming to rest, we can use the principles of kinematics. Let's break down the problem step by step:
1. Identify the given information:
- Initial velocity of the block, u = 2.00 m/s
- The incline angle, θ = 23.0°
- The block comes to rest, so the final velocity, v = 0 m/s
- The incline is frictionless, meaning there is no horizontal force acting on the block.
2. Analyze the motion along the incline:
- The component of the force due to gravity acting along the incline is mg * sin(θ), where m is the mass of the block and g is the acceleration due to gravity.
- The component of the force due to gravity perpendicular to the incline is mg * cos(θ), which cancels out with the normal force exerted by the incline.
3. Find the acceleration of the block:
- Since there is no horizontal force, the only force acting along the incline is the force due to gravity. Hence, F = m * g * sin(θ).
- According to Newton's second law, F = m * a, where a is the acceleration. Equating the two, we have m * g * sin(θ) = m * a.
- The mass, m, cancels out, so a = g * sin(θ).
4. Determine the distance traveled:
- To find the distance traveled up the incline, we need to calculate the time taken to come to rest.
- We can use the equation v = u + a * t, where v is the final velocity, u is the initial velocity, a is the acceleration, and t is the time taken to reach v.
- Since the final velocity is zero, the equation becomes 0 = u + a * t.
- Solving for t, we have t = -u / a.
- Note that we use the negative sign because the acceleration acts in the opposite direction to the initial velocity.
5. Calculate the distance traveled:
- The distance traveled, s, can be calculated using the equation s = u * t + (1/2) * a * t^2.
- Plugging in the values, s = u * (-u / a) + (1/2) * a * (-u / a)^2.
Let's substitute the given values and calculate the answer.
m g h = (1/2) m v^2
h = v^2/2g
h here is final height. I am not sure if that is what the question wants. If distance up the slope is required it is d = h /sin 23