A bolt is dropped from a bridge under construction, falling 81 m to the valley below the bridge. (a) How much time does it take to pass through the last 28 % of its fall? What is its speed (b) when it begins that last 28 % of its fall and (c) just before it reaches the ground?
To solve this problem, we can use the equations of motion for an object in free fall.
The equation for the time it takes for an object to fall from a certain height is given by:
h = (1/2) * g * t^2
Where:
h is the height
g is the acceleration due to gravity (approximately 9.8 m/s^2)
t is the time
First, let's calculate the time it takes to fall 81 m:
81 = (1/2) * 9.8 * t^2
t^2 = (81 * 2) / 9.8
t^2 = 16.53
t = √16.53
t ≈ 4.07 seconds
a) How much time does it take to pass through the last 28% of its fall?
To find the time it takes to pass through the last 28% of the fall, we need to calculate what 28% of 81 m is:
28% of 81 = (28/100) * 81
28% of 81 ≈ 22.68
To find the time, we'll subtract this distance from the initial height of the fall and use the same equation:
h = (1/2) * g * t^2
(81 - 22.68) = (1/2) * 9.8 * t^2
58.32 = 4.9 * t^2
t^2 = 58.32 / 4.9
t^2 = 11.9
t ≈ √11.9
t ≈ 3.44 seconds
Therefore, it takes approximately 3.44 seconds to pass through the last 28% of its fall.
b) What is its speed when it begins that last 28% of its fall?
To find the speed when it begins the last 28% of its fall, we can use the equation for velocity during free fall:
v = g * t
Where:
v is the velocity
g is the acceleration due to gravity (approximately 9.8 m/s^2)
t is the time
v = 9.8 * 3.44
v ≈ 33.7 m/s
Therefore, the speed when it begins the last 28% of its fall is approximately 33.7 m/s.
c) What is its speed just before it reaches the ground?
To find the speed just before it reaches the ground, we can use the same equation as before:
v = g * t
v = 9.8 * 4.07
v ≈ 39.9 m/s
Therefore, the speed just before it reaches the ground is approximately 39.9 m/s.
To answer these questions, we'll use the equations of motion to analyze the motion of the bolt as it falls from the bridge.
Let's start by calculating the time it takes to pass through the last 28% of its fall (a).
We can use the equation for the time taken (t) to fall from a given height (h) with respect to gravitational acceleration (g):
t = √(2h/g)
Since we are looking for the time taken to fall through only 28% of the total height, we first need to find that height.
28% of 81 m = 0.28 x 81 = 22.68 m
Now, we can calculate the time taken to fall through this height:
t = √(2h/g) = √[(2 x 22.68) / 9.8]
Calculating this expression will give us the answer to part (a).
Next, let's calculate the speed of the bolt when it begins the last 28% of its fall (b). We can use the equation for the final velocity (v) of an object in free fall:
v = √(2gh)
We already know the value of h (22.68 m). Rearranging the equation, we can find v.
Finally, let's find the speed of the bolt just before it reaches the ground (c). We can again use the equation for the final velocity (v). Since the bolt falls from rest at the bridge, the initial velocity (u) is 0.
v = √(2gh)
Now, you can plug in the values and calculate the required quantities. Note that the acceleration due to gravity (g) is approximately 9.8 m/s^2.
H=gt²/2
t =sqrt(2H/g) = sqrt(2•81/9.8)=4.07 s
h=(1-0.28)H =58.32 s
t₁ =sqrt(2h/g) =3.45 s.
t₂=4.07 – 3.45 = 0.62 s.
v₁=gt₁ =9.8•3.45 =33.81 m/s
v₂ = gt₂ =9.8•4.07 = 39.89 m/s