object thrown upward from an initial height of h0 feet with an initial velocity of v0 (in feet per second) is given by the formula
h(t)=−16t^2+v0t+h0 feet
where t is the amount of time in seconds after the ball was thrown. Also, the velocity of the object is given by
v(t)=−32t+v0. feet per second
When one uses the metric system, the equations become
h(t)=−4.9t^2+v0t+h0 meters
and
v(t)=−9.8t+v0 meters per second.
An object is projected upward from a height of 33 feet at a velocity of 97 feet per second.
Find the velocity of the object when it hits the ground.
oops. not so, due to h0
so, find t when h(t)=0
then use that to get v(t) at that time.
To find the velocity of the object when it hits the ground, we need to find the time it takes for the object to reach the ground first.
Given:
Initial height, h0 = 33 feet
Initial velocity, v0 = 97 feet per second
We know that when the object hits the ground, the height is 0. So, we can set h(t) = 0 and solve for t:
0 = -16t^2 + 97t + 33
To solve this quadratic equation, we can either factorize it or use the quadratic formula.
Using the quadratic formula:
t = (-b ± √(b^2 - 4ac)) / 2a
For our equation, a = -16, b = 97, and c = 33.
t = (-97 ± √(97^2 - 4(-16)(33))) / (2(-16))
Simplifying the equation:
t = (-97 ± √(9409 + 2112)) / (-32)
t = (-97 ± √11521) / (-32)
Now, finding the values of t by using both the positive and negative square root:
t₁ = (-97 + √11521) / (-32)
t₂ = (-97 - √11521) / (-32)
Since the object is going upwards initially, we only consider the positive value of time, t₁.
Now, let's calculate t₁:
t₁ = (-97 + √11521) / (-32)
Using a calculator, we find:
t₁ ≈ 6.52 seconds (rounded to two decimal places)
Now that we know the time it takes for the object to reach the ground is approximately 6.52 seconds, we can find the velocity at that time using the equation v(t) = -9.8t + v₀:
v(t) = -9.8 * 6.52 + 97
≈ -63.896 + 97
≈ 33.104 feet per second
Therefore, the velocity of the object when it hits the ground is approximately 33.104 feet per second.
To find the velocity of the object when it hits the ground, we need to determine the time it takes for the object to reach the ground first. We can do this by setting the height function equal to zero and solving for t.
Given:
h(t) = -16t^2 + v0t + h0
h(t) = 0 (since the object hits the ground)
h0 = 33 feet
v0 = 97 feet per second
Therefore:
-16t^2 + 97t + 33 = 0
To solve this quadratic equation, we can use factoring, completing the square, or the quadratic formula. In this case, let's use the quadratic formula:
t = (-b ± √(b^2 - 4ac)) / (2a)
Plugging in the values:
a = -16, b = 97, c = 33
t = (-97 ± √(97^2 - 4(-16)(33))) / (2(-16))
Now we can calculate the value of t.