A ball is thrown upward with a speed of 40 feet per second from the edge of a cliff 500 feet above the ground. What is the speed of the ball when it hits the ground? Use acceleration due to gravity as –32 feet per second squared and approximate your answer to 3 decimal places.
v₀ = 40 ft / s
h₀ = 500 ft
a = - 32 ft / s²
a = dv(t) / dt
dv(t) = a ∙ dt
v(t) = ∫ a ∙ dt
v(t) = ∫ - 32 ∙ dt
v(t) = - ∫ 32 ∙ dt
v(t) = - 32 t + v₀
v(t) = - 32 t + 40
v(t) = dh(t) / dt
dh(t) = v(t) ∙ dt
h(t) = ∫ v(t) ∙ dt
h(t) = ∫ ( - 32 t + 40) dt
h(t) = - ∫ 32 t dt + ∫ 40 dt
h(t) = - 32 ∫ t dt + ∫ 40 dt
h(t) = - 32 ∫ t² / 2 + ∫ 40 t + h₀
h(t) = - 16 t² + 40 t + h₀
h(t) = - 16 t² + 40 t + 500
The stone hits the ground when its height is 0:
- 16 t² + 40 t + 500 = 0
The solutions are:
t = ( 5 / 4 ) ( 1 - √ 21 ) = - 4.47822 sec
and
t = ( 5 / 4 ) ( 1 + √ 21 ) = 6.97822 sec
Time can't be negative so t = 6.97822 sec
v = | v(t) | = | - 32 t + 40 | =
| - 32 ∙ 6.978226 + 40 | =
| -183.303232 | = 183.303232 ft / s
v = 183.303 ft / s
approximately to 3 decimal places
Well, let's see... the ball is thrown upward with a speed of 40 feet per second, but unfortunately, gravity isn't too fond of that. It's like when you try to jump really high, but your pants get caught on a tree branch and bring you back down. Anyway, back to the ball. With gravity pulling it down at a rate of -32 feet per second squared, it's in for quite a ride.
To find the speed of the ball when it hits the ground, we have to take into account two things: the initial velocity and the acceleration due to gravity. The initial velocity is 40 feet per second, but remember, it's heading upward. So, let's call that a positive 40. Now, the acceleration due to gravity is -32 feet per second squared, because gravity always likes to bring things down.
To calculate the time it takes for the ball to hit the ground, we can use the formula: time = (final velocity - initial velocity) / acceleration. In this case, the final velocity would be 0, because the ball stops when it hits the ground. So, let's plug the numbers in:
time = (0 - 40) / -32
Using a little math magic, we get:
time = 1.25 seconds
Now that we know the time it takes for the ball to hit the ground, we can calculate the speed it reached. We can use the formula: speed = initial velocity + (acceleration * time). Plugging in the numbers:
speed = 40 + (-32 * 1.25)
Drumroll, please...
speed = -40 feet per second
So, the speed of the ball when it hits the ground is approximately -40 feet per second. I guess you could say it's falling with style!
To find the speed of the ball when it hits the ground, we need to determine the time it takes for the ball to reach the ground.
Using the kinematic equation:
v = u + at
where:
v = final velocity (which is 0 when the ball hits the ground)
u = initial velocity (40 feet per second)
a = acceleration (-32 feet per second squared)
0 = 40 - 32t
Rearranging the equation:
32t = 40
Solving for t:
t = 40 / 32
t ≈ 1.25 seconds
Now, to find the speed when it hits the ground, we can use the equation:
v = u + at
v = 40 + (-32 * 1.25)
Calculating:
v = 40 - 40
v = 0
Therefore, the speed of the ball when it hits the ground is 0 feet per second.
To find the speed of the ball when it hits the ground, we can use the concept of projectile motion and apply the equations of motion.
First, let's calculate the time it takes for the ball to reach the ground. We know that the initial velocity of the ball is 40 feet per second, the acceleration due to gravity is -32 feet per second squared, and the displacement is -500 feet (negative because the ball is moving upwards).
Using the equation of motion for displacement, velocity, and time:
d = ut + 1/2at^2
where:
d = displacement
u = initial velocity
a = acceleration
t = time
Substituting the known variables:
-500 = 40t + 1/2(-32)t^2
Rearranging the equation and solving for t, we get a quadratic equation:
16t^2 - 40t - 500 = 0
Solving this quadratic equation, we find that t ≈ 7.75 seconds (approximated to two decimal places).
Next, let's find the final velocity of the ball when it hits the ground. The final velocity can be calculated using the equation of motion:
v = u + at
where:
v = final velocity
u = initial velocity
a = acceleration
t = time
Substituting the known values:
v = 40 + (-32)*(7.75)
Calculating that, we get:
v ≈ -214.00 feet per second
Note that the negative sign indicates that the velocity is directed downwards.
To approximate the answer to three decimal places, we get:
v ≈ -214.000 feet per second
Therefore, when the ball hits the ground, its speed is approximately 214.000 feet per second.