An autographed baseball rolls off of a 1.3 m high desk and strikes the floor 0.74 m away from the desk. The acceleration of gravity is 9.81 m/s2 .
How fast was it rolling on the desk before it fell off?
1.44
To solve this problem, we can use the principles of projectile motion.
We are given the height of the desk (h) as 1.3 m and the horizontal distance the baseball rolls on the floor (d) as 0.74 m. The acceleration due to gravity (g) is 9.81 m/s².
To find the initial velocity (v₀) of the baseball, we can use the equations of motion:
1. The vertical displacement equation:
h = v₀² / (2 * g)
2. The horizontal displacement equation:
d = v₀ * t
Since the time (t) it takes for the ball to hit the floor is not provided, we can use the vertical displacement equation to obtain t.
Rearrange the vertical displacement equation to solve for v₀:
v₀ = √(2 * g * h)
Substitute the given values into the equation:
v₀ = √(2 * 9.81 * 1.3)
v₀ ≈ 5.03 m/s
Therefore, the initial velocity of the baseball before it fell off the desk was approximately 5.03 m/s.
To find the speed at which the baseball was rolling on the desk before it fell off, we can use the principle of conservation of energy.
Here's the step-by-step explanation:
1. First, let's calculate the gravitational potential energy of the ball when it was on the desk. The gravitational potential energy can be calculated using the equation:
Potential energy = mass × acceleration due to gravity × height
In this case, the height is 1.3 m, the acceleration due to gravity is 9.81 m/s^2, and the mass of the baseball is not given. Since we are only interested in the speed, we can ignore mass as it cancels out in the subsequent calculations.
Potential energy = 9.81 m/s^2 × 1.3 m = 12.7533 J (Joules)
2. Next, we can calculate the kinetic energy of the ball when it hits the ground. The kinetic energy can be calculated using the equation:
Kinetic energy = 0.5 × mass × velocity^2
However, since we don't have the mass of the baseball, we can use the fact that potential energy is equal to kinetic energy when the ball reaches the ground. So we can equate the potential energy (12.7533 J) to the kinetic energy and solve for velocity (speed).
12.7533 J = 0.5 × mass × velocity^2
3. Rearranging the equation, we get:
velocity^2 = (2 × potential energy) / mass
4. Now, we need to find the distance the ball rolled on the desk (0.74 m). This distance is related to the initial velocity of the ball before it fell off the desk. Using the equation for uniform acceleration:
distance = (initial velocity × time) + (0.5 × acceleration × time^2)
In this case, the acceleration is 0 (since the ball rolls without any acceleration on the desk) and the time is not provided. However, we can assume that the ball rolls on the desk for a very short time compared to the time it takes to fall, so we can ignore this term.
distance = initial velocity × time
5. Combining the equation for distance with the given values (distance = 0.74 m), we can solve for the initial velocity:
0.74 m = initial velocity × time
6. Since the time is unknown, we need another equation to solve for it. The total time it takes for the ball to fall can be calculated using the equation:
time = sqrt((2 × distance) / acceleration due to gravity)
Plugging in the values (distance = 1.3 m, acceleration = 9.81 m/s^2), we can find the total time it takes for the ball to fall.
7. Finally, we can substitute the value of time into the equation from Step 5 and solve for the initial velocity of the ball.
Note that the answer will depend on the mass of the baseball, which is not provided in the question.
Calculate the time it takes to fall H = 1.3 m.
t = sqrt (2H/g) = 0.515 s
Vx = 0.74 m/0.515s = ___
Vx does not change as the ball falls to the floor