An autographed baseball rolls off of a 0.59 m
high desk and strikes the floor 0.42 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
.
Answer in units of m/s
To find the initial speed of the autographed baseball before it fell off the desk, we can use the principles of projectile motion.
First, we need to understand the key variables involved in this problem:
- Initial vertical position (y0): Given as 0.59 m. This is the height from which the baseball falls.
- Horizontal distance (x): Given as 0.42 m. This is the horizontal distance between the desk and the point of impact with the floor.
- Acceleration due to gravity (g): Given as 9.81 m/s^2.
When the baseball falls off the desk, it will form a projectile, meaning it will have both horizontal (x) and vertical (y) components of motion. The horizontal motion (x) will be constant since there is no horizontal force acting on the baseball. The vertical motion (y) will be under the influence of gravity.
To find the initial speed of the baseball, we can split the problem into two parts:
1. Vertical motion:
Using the equation for vertical displacement:
y = y0 + v0y*t - (1/2)gt^2,
where:
- y0: Initial vertical position (given as 0.59 m)
- v0y: Initial vertical velocity (what we're trying to solve)
- t: time (unknown)
- g: acceleration due to gravity (given as 9.81 m/s^2)
At the maximum height of the projectile's vertical motion, the vertical velocity will be zero. So we can substitute y = 0 and solve for v0y.
0 = 0.59 + v0y*t - (1/2)gt^2
2. Horizontal motion:
Using the equation for horizontal displacement:
x = v0x*t,
where:
- x: Horizontal distance (given as 0.42 m)
- v0x: Initial horizontal velocity (what we're trying to solve)
- t: time (which we can find from the vertical motion)
Now, we can solve for t using the vertical motion equation and substitute it into the horizontal motion equation to find v0x.
1. Solve for t:
Since the baseball falls vertically, the time it takes to reach the floor can be found from the vertical motion equation when y = 0:
0 = 0.59 + v0y*t - (1/2)gt^2
Simplifying the equation yields a quadratic expression:
-0.5gt^2 + v0y*t + 0.59 = 0
We can solve this quadratic equation to find the value of t.
2. Substitute t into the horizontal motion equation:
Now that we have the value of t, we can substitute it into the horizontal motion equation to find v0x:
x = v0x*t
Solving for v0x gives us the initial horizontal velocity.
By solving the above equations, you can find the initial horizontal velocity.