A ball is thrown horizontally from the roof of a building 45cm tall and lands 24.0m from the base. What was the ball's initial velocity?
I'm going to assume you meant 45.0 m tall for the building (rather than cm).
Vertically (take upward as positive):
d = vit + (1/2)at ²
-45 = 0*t + (1/2)(-9.81)t ²
t ~ 3.0 s
Horizontally (take direction of throw as positive)
d = vit
24 = vi (3.0)
vi ~ 8 m/s
0.0m
To find the ball's initial velocity, we can use the horizontal motion equation:
d = v * t
Where:
d = horizontal distance travelled (24.0 m)
v = initial velocity of the ball
t = time of flight
However, before we can use this equation, we need to find the time of flight. In this case, since the ball is thrown horizontally, the vertical motion does not affect the horizontal motion. Therefore, the time taken to fall from the roof of the building is the same as the time taken to travel horizontally.
We can use the vertical motion equation:
h = (1/2) * g * t^2
Where:
h = height of the building (45 cm = 0.45 m)
g = acceleration due to gravity (assumed to be 9.8 m/s^2)
t = time of flight
Rearranging the equation to solve for time:
t^2 = (2h) / g
t = sqrt((2h) / g)
Substituting the given values:
t = sqrt((2 * 0.45) / 9.8)
t ≈ 0.0969 s
Now, we can use the horizontal motion equation to find the initial velocity:
24.0 m = v * 0.0969 s
v = 24.0 m / 0.0969 s
v ≈ 247.26 m/s
Therefore, the ball's initial velocity was approximately 247.26 m/s.
To find the initial velocity of the ball, we can use the equations of motion. In this scenario, the motion of the ball is influenced only by gravity in the vertical direction, since it is thrown horizontally.
Let's assume the initial velocity of the ball is denoted as 'v₀', the height of the building is 'h', and the horizontal distance traveled by the ball is 'd'.
The time taken for the ball to fall from the roof to the ground can be found by using the equation:
h = (1/2) * g * t²
Where 'g' is the acceleration due to gravity, approximately 9.8 m/s².
Rearranging the equation, we get:
t² = (2h) / g
Now, plugging in the values, we have:
t² = (2 * 0.45) / 9.8
t² ≈ 0.092
Next, we can calculate the time taken for the ball to travel the horizontal distance:
d = v₀ * t
Rearranging the equation, we have:
t = d / v₀
Substituting the value of 't' obtained earlier, we get:
0.092 = 24.0 / v₀
Solving for 'v₀', we find:
v₀ ≈ 24.0 / 0.092
v₀ ≈ 260.9 m/s
Therefore, the initial velocity of the ball is approximately 260.9 m/s.