An autographed baseball rolls off of a 0.59 m high desk and strikes the floor 0.10 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^

Answer in units of m/s.

To find the initial velocity of the baseball as it rolls off the desk, we can use the equation of motion:

d = v_i*t + (1/2)*a*t^2

Where:
d = distance traveled (0.10 m)
v_i = initial velocity (unknown)
a = acceleration due to gravity (-9.81 m/s^2, negative because it acts downward)
t = time it takes to fall (unknown)

We can solve for t using the equation:

d = v_i*t + (1/2)*a*t^2

Rearranging the equation:

0.10 = v_i*t + (1/2)*(-9.81)*t^2

Since the baseball is rolling horizontally on the desk before falling, the initial vertical velocity is zero (v_y = 0). Therefore, the time it takes to fall can be found using the equation:

d = (1/2)*a*t^2

Rearranging the equation:

0.59 = (1/2)*(-9.81)*t^2

Now we have a system of two equations which we can solve simultaneously. Solving the second equation for t^2:

t^2 = (2*0.59) / (-9.81)

t^2 = -0.120

Since time cannot be negative, we discard the negative value and take the positive square root:

t = √0.120

t = 0.346 s

We can now substitute the value of t into the first equation to solve for v_i:

0.10 = v_i*(0.346) + (1/2)*(-9.81)*(0.346)^2

0.10 = v_i*(0.346) - 0.050*(0.346)^2

Rearranging the equation:

v_i = (0.10 + 0.050*(0.346)^2) / (0.346)

v_i ≈ 1.345 m/s

Therefore, the baseball was rolling on the desk at approximately 1.345 m/s before it fell off.

To find the speed at which the autographed baseball was rolling on the desk before it fell off, we can use the principle of conservation of energy.

First, we need to determine the potential energy of the baseball when it was on the desk. The potential energy (PE) can be calculated using the equation:

PE = mgh

Where:
m = mass of the baseball (assumed to be constant)
g = acceleration due to gravity (9.81 m/s^2)
h = height of the desk (0.59 m)

Next, we can find the kinetic energy (KE) of the baseball just before it hits the ground. The kinetic energy can be calculated using the formula:

KE = (1/2)mv^2

Where:
m = mass of the baseball (assumed to be constant)
v = velocity/speed of the baseball just before it hits the ground

According to the principle of conservation of energy, the potential energy of the baseball on the desk is converted into its kinetic energy just before it hits the ground, assuming no other forms of energy loss. Therefore, we can equate the two equations:

PE = KE

mgh = (1/2)mv^2

We can cancel out the mass (m) from both sides of the equation:

gh = (1/2)v^2

Now, we can solve for v:

v^2 = 2gh

Taking the square root of both sides:

v = √(2gh)

Plugging in the given values of g (9.81 m/s^2) and h (0.59 m) into the equation:

v = √(2 * 9.81 m/s^2 * 0.59 m)

Calculating this expression, we get:

v ≈ √11.571 = 3.40 m/s

Therefore, the speed at which the autographed baseball was rolling on the desk before it fell off is approximately 3.40 m/s.