An autographed baseball rolls off of a 0.82 m
high desk and strikes the floor 0.28 m away
from the desk.
How fast was it rolling on the desk before
it fell off? The acceleration of gravity is
9.81 m/s
2
0.28 m = (horizontal velocity component)*(fall time)
The fall time is sqrt(2H/g) = 0.4089 s
H = 0.82 m
Solve for the horizontal velocity component
0.685 m/s
To find the initial speed of the rolling baseball before it fell off the desk, we can use the principles of projectile motion.
We can start by using the equation for horizontal motion:
Horizontal distance (d) = Initial horizontal velocity (Vx) × Time (t)
Given that the horizontal distance is 0.28 m, we know the horizontal velocity is the same as the final horizontal velocity when the ball hit the floor.
Next, we can use the equation for vertical motion:
Vertical distance (h) = Initial vertical velocity (Vy) × Time (t) + (0.5) × acceleration due to gravity (g) × Time (t)^2
Given that the vertical distance is 0.82 m, we know the initial vertical velocity is 0 since the ball was initially at rest on the desk. We also know the acceleration due to gravity is 9.81 m/s^2.
Since the time it takes for the ball to hit the floor is the same for both horizontal and vertical motion, we can equate the expressions for time in both equations:
(Vx × t) = 0.28 m
Vy × t + (0.5) × 9.81 m/s^2 × t^2 = 0.82 m
Solving the first equation for time (t), we get:
t = 0.28 m / Vx
Substituting this value of time into the second equation, we get:
0.28 m / Vx × Vy + (0.5) × 9.81 m/s^2 × (0.28 m / Vx)^2 = 0.82 m
Rearranging and simplifying the equation, we get:
0.14 × Vy / Vx + 0.0619 / Vx^2 = 0.82 / 9.81
Multiply both sides by Vx^2 to eliminate the denominators:
0.14 × Vy × Vx + 0.0619 = 0.082 × Vx
Rearranging the equation, we get:
0.14 × Vy × Vx = 0.082 × Vx - 0.0619
Dividing both sides by Vx:
0.14 × Vy = 0.082 - 0.0619 / Vx
Simplifying further:
0.14 × Vy = 0.0201 / Vx
Finally, divide both sides by Vy to isolate Vy:
Vy = 0.0201 / (0.14 × Vx)
Therefore, the initial vertical velocity of the rolling baseball is given by Vy = 0.0201 / (0.14 × Vx).
To determine the speed at which the autographed baseball was rolling on the desk before it fell off, we can use the principles of linear motion and conservation of energy.
First, let's consider the vertical motion of the baseball. Given that it rolls off a 0.82 m high desk, we can use the equation for vertical displacement:
Δy = (v₀y * t) + (0.5 * a * t²)
In this equation:
- Δy is the vertical displacement (0.82 m for this case)
- v₀y is the initial vertical velocity (which we want to find)
- t is the time of fall
- a is the acceleration due to gravity (-9.81 m/s²)
Since we are only interested in the initial velocity, we can ignore the second term on the right side of the equation.
Δy = v₀y * t
Next, let's consider the horizontal motion of the baseball. The horizontal distance it travels is given as 0.28 m.
We can determine the time of flight using the horizontal distance and the horizontal component of the initial velocity:
Δx = v₀x * t
Since the ball rolls off the desk, it does not have an initial horizontal velocity (v₀x = 0).
0.28 m = 0 * t
t = 0
From this, we can conclude that the time of flight is 0.
Now, let's go back to the equation for vertical displacement:
Δy = v₀y * t
Since t is 0, the vertical displacement is also 0 (Δy = 0). This means that the initial vertical velocity (v₀y) is also 0.
Therefore, the ball was not rolling vertically on the desk before it fell off. It was only rolling horizontally.
In conclusion, the ball was rolling horizontally on the desk before it fell off, and its vertical motion was not relevant to the calculation of its initial speed.