Stephanie hits a volleyball from a height of 0.80 m and gives it an initial velocity of 7.6 m/s straight up.
a) how high will the volleyball go?
b) how long will it take the ball to reach its maximum height?
the free-fall equation is ... h = 1/2 g t^2 + Vo t + Ho
... h = -4.9 t^2 + 7.6 t + 0.80
a) the time (t) for max height is on the axis of symmetry
... t = -b / 2a = -7.6 / -9.8
... calculate the time , and substitute into the free-fall equation for max height
b) you already found the time
a) Well, the volleyball seems to have quite the lofty ambitions! To determine how high it will go, we need to use a little physics magic. Since the volleyball is going straight up, we can treat it as a projectile in free fall. We'll need to figure out the time it takes for the ball to reach its maximum height.
b) Now, to find out the time it takes for the ball to reach its maximum height, we need to use a little equation called "Time of Flight Formula". With a starting velocity of 7.6 m/s and assuming there are no external forces acting on the ball, we can use the formula: time = (final velocity - initial velocity) / acceleration.
c) Let me do the calculations for you. Taking into account that the acceleration due to gravity is 9.8 m/s², we have:
time = (0 - 7.6) / (-9.8)
So, the time it takes for the ball to reach its maximum height is approximately 0.776 seconds.
d) Now that we've found the time it takes to reach the maximum height, we can use another formula to calculate how high the ball will go. This formula is known as the "Displacement Formula", and it goes like this: displacement = (initial velocity * time) + (1/2 * acceleration * time²).
e) By plugging in the values, we have: displacement = (7.6 * 0.776) + (0.5 * -9.8 * 0.776²).
f) After some quick math, the displacement of the ball turns out to be approximately 2.39 meters.
a) So, to answer your question, the volleyball will reach a height of approximately 2.39 meters.
b) And the time it takes for the ball to reach its maximum height is approximately 0.776 seconds. But remember, all of this calculation assumes no factors like air resistance or any real-life shenanigans!
To find the answers, we can use the equations of motion for vertical motion. We'll assume that air resistance is negligible.
a) To find how high the volleyball will go, we can use the following equation of motion:
v_f^2 = v_i^2 + 2 * a * d
Where:
- v_f is the final velocity (which will be 0 at the highest point as the ball changes direction)
- v_i is the initial velocity (7.6 m/s)
- a is the acceleration due to gravity (-9.8 m/s^2, since the ball is moving upwards)
- d is the displacement or height (which is what we want to find)
At the highest point, the final velocity will be 0 m/s. Rearranging the equation, we have:
d = (v_f^2 - v_i^2) / (2 * a)
Substituting the given values, we can calculate:
d = (0 - (7.6)^2) / (2 * (-9.8))
Calculating the value, we find:
d ≈ 2.90 m
Therefore, the volleyball will go approximately 2.90 m high.
b) To find the time it takes for the ball to reach its maximum height, we can use the following equation of motion:
v_f = v_i + a * t
Where:
- v_f is the final velocity (0 m/s, at the highest point)
- v_i is the initial velocity (7.6 m/s)
- a is the acceleration due to gravity (-9.8 m/s^2, since the ball is moving upwards)
- t is the time (which is what we want to find)
Rearranging the equation, we have:
t = (v_f - v_i) / a
Substituting the given values, we can calculate:
t = (0 - 7.6) / (-9.8)
Calculating the value, we find:
t ≈ 0.78 s
Therefore, the ball will take approximately 0.78 seconds to reach its maximum height.
To answer these questions, we need to consider the principles of projectile motion. Let's break down the problem and determine the steps to find the answers:
a) To determine how high the volleyball will go, we need to find the maximum height reached by the projectile. We can use the formula:
vf^2 = vi^2 + 2ad
where vf is the final velocity (zero when the ball reaches its highest point), vi is the initial velocity (7.6 m/s), a is the acceleration due to gravity (-9.8 m/s^2), and d is the displacement (the maximum height). Rearranging the formula, we have:
d = (vf^2 - vi^2) / (2a)
Using this formula, plug in the given values to calculate the height the volleyball will reach.
b) To find the time taken to reach the maximum height, we can use the formula for the time of flight:
t = (vf - vi) / a
Where t is the time taken, vf is the final velocity (zero at the highest point), vi is the initial velocity (7.6 m/s), and a is the acceleration due to gravity (-9.8 m/s^2).
Now, let's calculate the answers!
a) Substitute the given values into the formula:
d = (0 - 7.6^2) / (2 * -9.8)
Calculate the square of 7.6 first:
d = (-57.76) / -19.6
Divide:
d ≈ 2.95 m
Therefore, the volleyball will reach a height of approximately 2.95 meters.
b) Substitute the given values into the formula:
t = (0 - 7.6) / -9.8
Calculate:
t ≈ 0.78 s
Therefore, it will take approximately 0.78 seconds for the ball to reach its maximum height.
So, the answers are:
a) The volleyball will go approximately 2.95 meters high.
b) It will take approximately 0.78 seconds to reach the maximum height.