In Pittsburgh, children bring their old jack-o-lanterns to the top of the Carnegie Science Center for the annual Pumpkin Smash and compete for accuracy in hitting a target on the ground. Suppose that the building is 23m high and that the bulls-eye is a horizontal distance of 4.5m from the launch point. If the pumpkin is thrown horizontally, what is the launch speed needed to hit the bulls-eye?
To determine the launch speed needed to hit the bulls-eye, we can use the following steps:
Step 1: Identify the known variables and parameters:
- Height of the Carnegie Science Center building (h) = 23m
- Horizontal distance to the bulls-eye (d) = 4.5m
Step 2: Understand the motion of the pumpkin:
Since the pumpkin is thrown horizontally, its initial vertical velocity will be zero. This means the only force acting on it will be gravity, causing it to fall in a parabolic trajectory. Therefore, we can use the equations of motion to find the launch speed.
Step 3: Determine the time of flight:
To calculate the time taken by the pumpkin to fall from the top of the building to the ground, we can use the equation:
h = (1/2) * g * t^2
where g is the acceleration due to gravity (approximately 9.8 m/s^2) and t is the time of flight.
Simplifying the equation, we find:
t^2 = (2h) / g
t = √((2h) / g)
Substituting the given values:
t = √((2 * 23) / 9.8) ≈ 2.418 seconds
Step 4: Calculate the launch speed:
Now that we have the time of flight, we can find the launch speed using the horizontal distance traveled by the pumpkin:
d = v * t
where d is the horizontal distance, v is the launch speed, and t is the time of flight.
Solving for the launch speed:
v = d / t
v = 4.5 / 2.418 ≈ 1.860 m/s
Therefore, to hit the bulls-eye, the launch speed needed for the pumpkin is approximately 1.860 m/s.
how long does it take to fall 23m?
4.9t^2 = 23
speed = distance/time