A block is given an initial velocity of 7.00 m/s up a frictionless 19.0° incline. How far up the incline does the block slide before coming (momentarily) to rest?
It will rise a vertical distance H until the potential energy increase MgH equals the initial kinetic energy,
(1/2) M V^2
Threfore H = V^2/(2 g)
The distance X that it moves up the incline is given by
sin 19 = H/X
To find how far up the incline the block slides before coming to rest, we can use the equations of motion. The key here is that the block comes to rest, meaning its final velocity is 0.
The block’s motion can be divided into two components: one parallel to the incline (up the incline) and one perpendicular to the incline. We only need to consider the component parallel to the incline since there is no friction acting on the block.
Let’s break down the problem step by step:
Step 1: Calculate the acceleration of the block
Since there is no friction, the only force acting on the block parallel to the incline is the component of its weight. We can calculate this force as follows:
Force parallel to the incline = mg * sin(theta)
where m is the mass of the block and theta is the angle of the incline. In this case, the angle is 19.0°.
Step 2: Find the time it takes for the block to stop
Using the equation of motion:
final velocity = initial velocity + (acceleration * time)
We know the final velocity is 0, the initial velocity is 7.00 m/s, and the acceleration is calculated in step 1. Solving for time will give us the duration it takes for the block to come to rest.
Step 3: Calculate the distance traveled up the incline
Using the equation of motion:
distance = (initial velocity * time) + (0.5 * acceleration * time^2)
Substituting the values we have, which are the initial velocity from the problem and the time we calculated in step 2, we can solve for the distance traveled up the incline.
Once you plug in the numbers and work through the calculations, the final result will give you the distance up the incline that the block slides before coming to rest.