My question is
A block is given an intial speed of 4.0 s^-1 m up the 22 degree plane
How far up the plane will it go?
How much time elapses before it returns to its starting point? Ignore friction.
Ok my teacher said use this to find acceleration a = g sin è and then I know where to go from there but were does this formula come from?
g is downward. Since the block cannot slid downward, it can only slide along the surface. So the component of g in the direction of the surface is gsin22.
http://www.lastufka.net/lab/cars/why/ramps.gif
Notice the two components: down the ramp, mgsintheta, and the other component of g, perpendicular, mgcosTheta. That perpendicular force assists friction, but in this case we are ignoring it.
The formula you mentioned, a = g sin è, actually comes from resolving the force components acting on an object placed on an inclined plane. Let's break it down step by step:
1. Forces acting on the block:
When the block is on the inclined plane, two main forces act upon it: the gravitational force (mg) directed vertically downwards and the normal force (N) perpendicular to the plane. These two forces can be resolved into their components.
2. Resolution of forces:
The gravitational force (mg) can be broken down into two components:
- mg sin è: This component acts parallel to the inclined plane, in the upward direction.
- mg cos è: This component acts perpendicular to the inclined plane and balances the normal force.
3. Acceleration:
The net force acting on the block parallel to the plane is given by F = mg sin è. According to Newton's second law (F = ma), the acceleration (a) of the block parallel to the plane is thus given by a = g sin è.
4. Applying the formula:
To find out how far the block will go up the plane, we need to use the equation of motion that relates distance (d), initial speed (u), time (t), and acceleration (a):
d = ut + (1/2)at^2
Since the initial speed (u) is given as 4.0 m/s and the acceleration (a) is g sin è, we can now solve for the distance (d) traveled up the plane.
5. Returning to starting point:
To find the time it takes for the block to return to its starting point, we need to consider that the vertical displacement must be zero. This means that the vertical component of the block's velocity becomes zero again. We can use the equation of motion for the vertical direction to find the time (t) it takes to reach this point.
Final thoughts:
By using the given initial speed, the angle of the plane, the formula for acceleration on an inclined plane (a = g sin è), and the equations of motion, you can find both the distance traveled up the plane and the time it takes for the block to return to its starting point.