if a ball is thrown downward from a roof top 100meters tall and hits the ground in 3 seconds how hard is it thrown?
d = 1/2 g t^2 + Vo t ... 100 = 1/2 * 9.8 * 3^2 + 3 Vo
hf=ho+v*t-1/2 g t^2
hf= final position
ho=initial position
v= intial velocity, upward is +
g = -9.8m/s^2
t is time in flight
0=100+v*3+1/2 * -9.8*9
-100=3v-4.9*9
solve for v. I get about -18.6 m/s (negative means downward)
To determine how hard the ball is thrown, we need to calculate its initial velocity (or speed). We can use the equation of motion:
\( s = ut + \frac{1}{2}gt^2 \),
where:
- \( s \) is the displacement (in this case, the distance fallen)
- \( u \) is the initial velocity
- \( t \) is the time taken
- \( g \) is the acceleration due to gravity (\( 9.8 \, \text{m/s}^2 \)).
In this case, the ball is falling downward, so the displacement is 100 meters (negative value). The time taken is 3 seconds, and the acceleration due to gravity is \( -9.8 \, \text{m/s}^2 \) since the ball is moving downward. We can rewrite the equation as:
\( -100 = u(3) + \frac{1}{2}(-9.8)(3)^2 \).
Simplifying:
\( -100 = 3u - 44.1 \).
Rearranging the equation:
\( 3u = -100 + 44.1 \),
\( 3u = -55.9 \),
\( u = -\frac{55.9}{3} \).
Thus, the ball is thrown downward with an initial velocity of approximately \( -18.63 \, \text{m/s} \).
To determine how hard the ball is thrown, we need to calculate the initial velocity at which it was thrown. We can use the equation of motion for objects in free-fall to find the initial velocity.
Let's start by identifying the known variables:
- Initial height (h) = 100 meters
- Time taken (t) = 3 seconds
- Acceleration due to gravity (g) = 9.8 m/s^2 (assuming no air resistance)
The equation we can use is:
h = (1/2) * g * t^2 + v0 * t
Where:
- h is the initial height (100 meters)
- g is the acceleration due to gravity (9.8 m/s^2)
- t is the time taken (3 seconds)
- v0 is the initial velocity (what we want to find)
Rearranging the equation to solve for v0:
v0 = (h - (1/2) * g * t^2) / t
Now, let's substitute the values and calculate:
v0 = (100 - (0.5 * 9.8 * 3^2)) / 3
= (100 - (0.5 * 9.8 * 9)) / 3
= (100 - 44.1) / 3
= 55.9 / 3
≈ 18.63 m/s
Therefore, the ball is thrown with an initial velocity of approximately 18.63 meters per second downward from the rooftop.