. A tennis player serves the ball at a height of h with an initial velocity of 40m/s at
an angle of 4 0 with the horizontal. Knowing that the ball clears the 0.941m net by
152mm. Determine (a) the height h (b) the distance d from the net to where the ball
will land.
Well, well, well... looks like our tennis player is aiming high and clearing the net with quite a margin! Let's dive into these calculations and have some fun, shall we?
(a) To determine the height h, we need to find the vertical component of the initial velocity. Using a bit of trigonometry, we can calculate h as follows:
sin(40°) = h/40m/s
And now, let's solve for h:
h = 40m/s * sin(40°)
Now, I know math can be a bit serious, but we can always add a pinch of humor to it! So, let me grab my calculator and crunch the numbers...
*clownishly punching calculator buttons*
Ah, it seems that h is approximately 25.8 meters. So, our tennis player is truly soaring high!
(b) Now, let's determine the distance d from the net to where the ball will land. To do this, we need to find the horizontal component of the initial velocity. Again, employing some trigonometric wizardry, we can find d:
cos(40°) = d/40m/s
And now, the moment of truth!
d = 40m/s * cos(40°)
Let me calculate that for you...
*punching calculator in a clownish manner*
Voila! It appears that d is approximately 30.7 meters. So, our ball will land at a playful distance from the net.
I hope that brings a smile to your face! Remember, a little humor never hurts, even in math!
To determine the height h and the distance d, we can use the equations of projectile motion.
Step 1: Breaking down the initial velocity:
The initial velocity can be divided into horizontal and vertical components.
Given that the initial velocity (v₀) is 40 m/s and the launch angle (θ) is 40°, we can find the horizontal (v₀ₓ) and vertical (v₀ᵧ) components.
v₀ₓ = v₀ * cos(θ)
v₀ᵧ = v₀ * sin(θ)
Substituting the given values:
v₀ₓ = 40 m/s * cos(40°)
v₀ᵧ = 40 m/s * sin(40°)
Step 2: Determining the time of flight:
The time it takes for the ball to reach the highest point is the same as the time it takes to fall back down to the ground.
Using the vertical component equation:
h = v₀ᵧ * t - (1/2) * g * t², where g is the acceleration due to gravity (9.8 m/s²)
Since the ball's height is at its maximum at this point, the height is given as h = h + 0.941 m + 0.152 m = h + 1.093 m
Rearranging the equation:
0 = v₀ᵧ * t - (1/2) * g * t² - h
We can solve this quadratic equation for t.
Step 3: Calculating the time of flight:
Solving the quadratic equation using the quadratic formula:
t = (-b ± sqrt(b² - 4ac)) / (2a)
Where a = -0.5g, b = v₀ᵧ, and c = -h.
Substituting the given values and solving for t:
a = -0.5 * 9.8 m/s² = -4.9 m/s²
b = v₀ᵧ
c = -h - 1.093 m
t = (-v₀ᵧ ± sqrt(v₀ᵧ² - 4 * (-4.9) * (-h - 1.093))) / (2 * (-4.9))
Step 4: Determining the time taken to reach the net:
To determine how long it takes for the ball to reach the net, we need to find the time when the vertical position is equal to the net height.
Using the equation for height:
h = v₀ᵧ * t - (1/2) * g * t²
Since the height is given as 0.941 m, we can solve for t:
0.941 m = v₀ᵧ * t - (1/2) * g * t²
Now, find the positive time t when the ball reaches the net.
Step 5: Calculating the distance:
The horizontal distance d can be calculated using the horizontal component equation:
d = v₀ₓ * t
Substitute the values of v₀ₓ calculated in Step 1 and t determined in Step 4 to find the distance d from the net where the ball will land.
By following these steps, you can determine the height h and the distance d from the net to where the ball will land.
To solve this problem, we can use the principles of projectile motion. First, let's break down the given information:
Given:
- Initial velocity (vi) = 40 m/s
- Launch angle (θ) = 40 degrees
- The net height (h_net) = 0.941 m
- The ball clears the net by 152 mm (0.152 m)
(a) To determine the height (h) at which the tennis player serves the ball, we need to find the vertical component of the initial velocity. We can use trigonometry to find:
Vertical component of velocity (vi_y) = vi * sin(θ)
Using the given values, we have:
vi_y = 40 m/s * sin(40°)
Now, let's calculate vi_y:
vi_y = 40 m/s * sin(40°) ≈ 25.65 m/s
Therefore, the height at which the tennis player serves the ball is approximately 25.65 meters.
(b) To find the distance (d) from the net to where the ball will land, we need to consider the horizontal component of the initial velocity. The time of flight can be calculated using the vertical component of velocity and the acceleration due to gravity (g = 9.8 m/s²):
Time of flight (t) = (2 * vi_y) / g
Substituting the values:
t = (2 * 25.65 m/s) / 9.8 m/s² ≈ 5.21 s
Now, we can calculate the horizontal distance using the formula:
Horizontal distance (d) = vi * cos(θ) * t
Using the given values, we have:
d = 40 m/s * cos(40°) * 5.21 s
Now, let's calculate d:
d = 40 m/s * cos(40°) * 5.21 s ≈ 161.88 m
Therefore, the distance from the net to where the ball will land is approximately 161.88 meters.