You toss a tennis ball straight upward. At the moment it leaves your hand it is at a height of 1.5 m above the ground, and it is moving at a speed of 7.8 m/s.
(a) How much time does it take for the tennis ball to reach its maximum height?
(b) What is the maximum height above the ground that the tennis ball reaches?
(c) When the tennis ball is at a height of 2.2 m above the ground, what is its speed?
(d) When the ball hits the ground what is its velocity?
a. V = Vo + g*t.
V = 0 @ max h.
Vo = 7.8 m/s.
g = -9.8 m/s^2.
t = ?
b. h = 1.5 + Vo*t + 0.5g*t^2.
Vo = 7.8 m/s.
t = Value calculated in part a.
g = -9.8 m/s^2.
h = ?
c. V^2 = Vo^2 + 2g*d.
Vo = 7.8 m/s.
g = -9.8 m/s^2.
d = 2.2-1.5 = 0.7 m.
V = ?.
d. V^2 = Vo^2 + 2g*h.
Vo = 0 @ max h.
g = +9.8 m/s^2.
h = value calculated in part b.
V = ?.
(a) How much time does it take for the tennis ball to reach its maximum height?
Oh, I can already hear the tennis ball telling its friends, "Hey guys, I'm going to my maximum height, can't hang out right now!"
Anyway, let's find out how long it takes for the ball to reach its highest point. We can use the equation for motion in the vertical direction:
v = u + at
Where:
v = final velocity (which is 0 at the maximum height because it will change direction)
u = initial velocity (7.8 m/s upwards)
a = acceleration (which is equal to the acceleration due to gravity, approximately -9.8 m/s^2)
t = time
Plugging in the values, we have:
0 = 7.8 - 9.8t
Now, solving for t:
9.8t = 7.8
t = 7.8 / 9.8
So, it will take approximately 0.796 seconds for the tennis ball to reach its maximum height.
(b) What is the maximum height above the ground that the tennis ball reaches?
Well, if the tennis ball is going up, up, and away, it will eventually reach a point where it feels like it's floating. That's the maximum height! We can use the equation:
s = ut + 1/2 at^2
Where:
s = displacement (maximum height above the ground)
u = initial velocity (7.8 m/s upwards)
a = acceleration (which is equal to the acceleration due to gravity, approximately -9.8 m/s^2)
t = time (0.796 seconds)
Let's substitute the values:
s = 7.8(0.796) + 1/2(-9.8)(0.796)^2
s = 6.2076 - 3.1092
The maximum height of the tennis ball is approximately 3.0984 meters above the ground. It's like a little gravity-defying acrobat!
(c) When the tennis ball is at a height of 2.2 m above the ground, what is its speed?
Well, now the tennis ball is feeling extra bouncy at a height of 2.2 meters! Let's find out how speedy it is at this point. To do that, we'll need to use another equation:
v^2 = u^2 + 2as
Where:
v = final velocity (what we're trying to find)
u = initial velocity (7.8 m/s upwards)
a = acceleration (which is equal to the acceleration due to gravity, approximately -9.8 m/s^2)
s = displacement (2.2 meters)
Let's plug in the numbers and solve for v:
v^2 = (7.8)^2 + 2(-9.8)(2.2)
v^2 = 60.84 - 42.12
v^2 = 18.72
v ≈ √18.72
v ≈ 4.329
So, when the tennis ball is at a height of 2.2 meters above the ground, its speed is approximately 4.33 m/s. Watch out for those fast-flying tennis balls!
(d) When the ball hits the ground, what is its velocity?
Oh boy, now the tennis ball is coming down to meet the ground! Let's find out how fast it's going to be. Since we know that the initial velocity is 7.8 m/s upwards, we just need to flip the sign on the velocity to indicate it's moving downwards.
So, when the ball hits the ground, its velocity will be approximately -7.8 m/s. That's the sound of a very disappointed tennis ball saying, "I guess the ground just doesn't bounce back the same way!"
To answer these questions, we can use the equations of motion and the principles of projectile motion. Let's break down each question step-by-step:
(a) How much time does it take for the tennis ball to reach its maximum height?
To determine the time taken to reach the maximum height, we can use the kinematic equation:
vf = vi + a*t
Since the ball is moving upward, the final velocity (vf) at its maximum height is zero. The initial velocity (vi) is 7.8 m/s. The acceleration (a) due to gravity is -9.8 m/s^2 (negative because it acts in the opposite direction of motion). Plugging these values into the equation, we get:
0 = 7.8 - 9.8*t
Solving for t:
9.8*t = 7.8
t = 7.8 / 9.8
t ≈ 0.796 seconds
So, it takes approximately 0.796 seconds for the tennis ball to reach its maximum height.
(b) What is the maximum height above the ground that the tennis ball reaches?
To find the maximum height, we need to use the formula:
h = vi*t + (1/2)*a*t^2
Since the ball's initial vertical velocity (vi) is 7.8 m/s, and the time (t) is 0.796 seconds, we can substitute these values along with acceleration (a) into the equation:
h = 7.8 * 0.796 + (1/2) * (-9.8) * (0.796)^2
h ≈ 3.13 meters
So, the maximum height above the ground that the tennis ball reaches is approximately 3.13 meters.
(c) When the tennis ball is at a height of 2.2 m above the ground, what is its speed?
To find the speed of the tennis ball when it is at a certain height, we can use the following equation:
vf^2 = vi^2 + 2*a*h
Here, vi is the initial velocity, vf is the final velocity (which is what we're solving for), a is the acceleration, and h is the height.
Let's plug in the given values:
vi = 7.8 m/s
a = -9.8 m/s^2
h = 2.2 m
vf^2 = (7.8)^2 + 2*(-9.8)*2.2
vf^2 ≈ 60.84
vf ≈ √60.84
vf ≈ 7.80 m/s
So, when the tennis ball is at a height of 2.2 m above the ground, its speed is approximately 7.80 m/s.
(d) When the ball hits the ground, what is its velocity?
When the ball hits the ground, its velocity will be the negative of its initial velocity (vi), which is -7.8 m/s. The negative sign indicates the direction is opposite to its initial motion.
So, when the ball hits the ground, its velocity will be approximately -7.8 m/s.
To solve these problems, we can use the equations of motion for vertical motion under constant acceleration.
(a) The time it takes for the tennis ball to reach its maximum height can be found using the equation:
vf = vi + at
In this case, the final velocity (vf) is 0 m/s at the maximum height, the initial velocity (vi) is 7.8 m/s, and the acceleration (a) is the acceleration due to gravity, which is approximately -9.8 m/s^2 (taking downwards as negative).
0 = 7.8 - 9.8t
Solving this equation will give us the time it takes for the tennis ball to reach its maximum height.
(b) The maximum height above the ground that the tennis ball reaches can be determined using the equation:
Δy = vi*t + (1/2) * a * t^2
In this case, the initial vertical velocity (vi) is 7.8 m/s, the time (t) can be obtained from part (a), and the acceleration (a) is -9.8 m/s^2.
Substituting the given values into the equation will allow us to calculate the maximum height.
(c) To find the speed of the tennis ball when it is at a height of 2.2 m above the ground, we need to use the following equation:
vf^2 = vi^2 + 2 * a * Δy
In this case, the initial vertical velocity (vi) is 7.8 m/s, the acceleration (a) is -9.8 m/s^2, and the displacement (Δy) is the difference between the final height and the initial height (2.2 m - 1.5 m).
Solving this equation will give us the speed of the tennis ball.
(d) When the ball hits the ground, its velocity can be calculated using the equation:
vf = vi + at
In this case, the initial velocity (vi) is the speed of the tennis ball just before hitting the ground (which we calculated in part (d)), acceleration (a) is the acceleration due to gravity (-9.8 m/s^2), and final velocity (vf) would be the value we are trying to find.
Solving this equation will give us the velocity of the tennis ball when it hits the ground.
A). 1.2 s
B). 3.81 m
c).v= 6.8 m/s
d) v= 27.05 m/s