A ball rolls down a ramp that's set on a table, and from there falls off the table. The height of the ramp, length of the ramp, and height from the table are known, and the task is to find the horizontal displacement.
One can use mgh = 1/2mv^2 + 1/2kmv^2 to find the velocity at the bottom of the ramp, but from there, how does the downward angle affect the calculations for horizontal displacement? When assuming the speed is solely horizontal and breaking it into components (x = vt and y = 1/2gt^2, finding t, then using that to find x), the predicted x displacement is far too large, and when trying to factor in the angle the ball was rolling from, the displacements are still too far from the experimental value. How would one find the horizontal displacement correctly, when taking into account how the ball is being "launched" from the table with a known velocity that is not solely horizontal? Any ideas would be helpful! Attached is a picture to illustrate the situation.
You are way over thinking this. It leaves the table at some height h, at a horizontal velocity v.
The question is how long does it take to fall to the floor?
hf=hi+vi*t-1/2 g t^2 but vi in the horizontal direction is zero, so you can solve for time t to tall the distance h.
Now, knowing the time in the air, you calcualte
distance=horizontalvelocity*timeinair where horizontal velocity is the velociyt it left the table with originally: that does not change.
To find the horizontal displacement correctly, taking into account the angle at which the ball is launched from the table, you can use the following steps:
1. Start by breaking down the initial velocity of the ball into its horizontal and vertical components. Let's call the initial velocity of the ball "v" and the launch angle "θ". The horizontal component of the velocity is given by v₀x = v * cos(θ), and the vertical component is given by v₀y = v * sin(θ).
2. Use the vertical component of the velocity to find the time it takes for the ball to reach the ground after falling off the table. Since the ball falls freely under gravity, you can use the equation y = 1/2 * g * t², where "y" is the vertical displacement, "g" is the acceleration due to gravity, and "t" is the time. In this case, the vertical displacement is the height from the table. Solve for "t" in this equation.
3. Now that you have the time it takes for the ball to fall to the ground, you can use the horizontal component of the velocity to find the horizontal displacement. Since there are no horizontal forces acting on the ball after it falls off the table, the horizontal velocity remains constant. Therefore, the horizontal displacement is given by x = v₀x * t.
By following these steps, you can accurately find the horizontal displacement of the ball. Just make sure to use the correct values for the initial launch angle and velocities when doing the calculations.