A ball is rolling along a horizontal table top at a constant speed. The table top is 75.0 cm above horizontal floor. The ball rolls over the edge of the table and hits the floor below at a horizontal distance from the table edge of 1.06 m. You may ignore air resistance in the motion of the ball through the air.
(a) After leaving the table top, how long does it take for the ball to hit the floor?
(b) What was the initial speed of the ball?
(c) What is the final speed of the ball just before it hits the floor?
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I tried to solve for it.
time = 0.39s
initial velocity = 0
final velocity = 3.82 m/s
Can anyone please check? Thank you!
To solve this problem, we can use the principles of projectile motion and apply the equations of motion. Let's consider the motion of the ball in two dimensions: horizontal and vertical.
(a) To find the time it takes for the ball to hit the floor, we need to calculate the time it takes to reach the horizontal distance of 1.06 m from the table edge. We can use the horizontal motion equation for this:
Horizontal distance traveled (x) = initial horizontal velocity (vx) * time (t)
Given that the horizontal distance is 1.06 m and the ball is rolling at a constant speed, we can assume no horizontal acceleration. Therefore, the initial horizontal velocity (vx) remains constant throughout the motion. Hence, the equation becomes:
1.06 m = vx * t
To find the time, we need to know the horizontal velocity. Let's move on to finding that next.
(b) To determine the initial speed of the ball, we need to consider the vertical motion of the ball. Since the ball starts from rest vertically, its initial vertical velocity (vy) is zero. The vertical motion equation we can use is:
Vertical distance traveled (y) = initial vertical velocity (vy) * time (t) + (1/2) * acceleration due to gravity (g) * time^2
Given that the initial vertical velocity is zero, the equation simplifies to:
y = (1/2) * g * t^2
The vertical distance traveled is the initial height of the table, which is 75.0 cm (or 0.75 m). Therefore, the equation becomes:
0.75 m = (1/2) * g * t^2
We can now solve this equation for time (t) using the value of acceleration due to gravity (g = 9.8 m/s^2). Once we know the time, we can use it to calculate the initial horizontal velocity and answer (a).
(c) To find the final speed of the ball just before it hits the floor, we can use the vertical motion equation again. In the final moment before hitting the floor, the vertical velocity of the ball (vf) will be the final speed we're looking for. Using the equation:
vf = vy + g * t
Here, the initial vertical velocity (vy) is still zero and we now know the value of time (t) calculated in part (b). Substituting these values in, we can find the final speed of the ball as it hits the floor.
To summarize:
(a) Find time (t) using the horizontal motion equation: 1.06 m = vx * t
(b) Solve for the initial speed of the ball: 0.75 m = (1/2) * g * t^2 and vx = 1.06 m / t
(c) Calculate the final speed (vf) using vf = vy + g * t. Recall that vy = 0.