6) Mythreyan, a baseball player leads off the game and hits a long home run. The ball leaves the bat at an angle of 70.0o from the horizontal with a velocity of 55.0 m/s.
a. How far will it travel in the air?
b. What is its maximum height?
c. How much time does it spend in the air?
the vertical speed v = 55 sin70° - 9.8t
find t when v=0. That is when the max height occurs
(A) 55 cos70° * t
(B) if hit at height h, then y = h + (55 sin70°)*t - 4.9t^2
(C) solve for y=0
A) use 2 t, it also comes down from the peak.
C) is 2 t assuming batter is short
a. Well, if the ball leaves the bat at a 70.0o angle, it's definitely out to go on an adventure! With a velocity of 55.0 m/s, it's going to put some serious distance between itself and the stadium. So, the ball will travel a pretty far distance in the air. Let's hope it doesn't get lost along the way!
b. Now, let's talk about the ball's maximum height. You see, when a ball is hit at an angle, it goes up, up, and away before eventually coming back down. So, at some point, this ball is going to reach its highest point in the sky like a bird on its way to a record height. That's its maximum height!
c. As for how much time the ball spends in the air, that's a tricky question. The ball has to come down eventually, right? So, it will spend some time soaring through the air before gravity gives it a friendly reminder that what goes up must come down. Think of it as the ball's vacation time in the air before returning to reality.
To answer these questions, we can use the equations of motion for projectile motion. Let's break it down step-by-step:
Step 1: Convert the angle to radians
The angle is given in degrees, but we need to convert it to radians for calculations. To convert from degrees to radians, we use the formula: radians = degrees * (π/180).
a. Distance traveled in the air:
Step 2: Divide the given angle by 2 to get the launch angle.
70° / 2 = 35°
Step 3: Convert the launch angle to radians.
35° * (π/180) = 0.6109 radians
Step 4: Use the equation for horizontal distance traveled:
Range = (v^2 * sin(2θ)) / g
Where v is the initial velocity, θ is the launch angle, and g is the acceleration due to gravity (approximately 9.8 m/s^2).
Given:
v = 55.0 m/s
θ = 0.6109 radians
g = 9.8 m/s^2
Range = (55.0^2 * sin(2 * 0.6109)) / 9.8
Range ≈ 249.65 meters
Therefore, the ball will travel approximately 249.65 meters in the air.
b. Maximum height:
Step 5: Use the equation for maximum height:
h = (v^2 * sin^2(θ)) / (2 * g)
Given:
v = 55.0 m/s
θ = 0.6109 radians
g = 9.8 m/s^2
h = (55.0^2 * sin^2(0.6109)) / (2 * 9.8)
h ≈ 44.68 meters
Therefore, the ball will reach a maximum height of approximately 44.68 meters.
c. Time spent in the air:
Step 6: Use the equation for time of flight:
Time = (2 * v * sin(θ)) / g
Given:
v = 55.0 m/s
θ = 0.6109 radians
g = 9.8 m/s^2
Time = (2 * 55.0 * sin(0.6109)) / 9.8
Time ≈ 6.39 seconds
Therefore, the ball will spend approximately 6.39 seconds in the air.
To calculate the answers to these questions, we can use the basic equations of projectile motion. Let's break down each question and determine the steps to find the answers.
a. To find how far the ball will travel in the air, we need to calculate the horizontal distance covered by the ball. Since there is no horizontal acceleration, the horizontal velocity remains constant throughout the motion. We can calculate the horizontal distance using the formula:
Horizontal distance = horizontal velocity × time
To find the time, we need to calculate the time of flight, which is the total time the ball spends in the air. We can use the equation:
Time of flight = (2 × vertical velocity) / acceleration due to gravity (or) √(2 × height / g) (where g is the acceleration due to gravity, approximately 9.8 m/s²)
Once we have the time of flight, we can calculate the horizontal distance using the equation above.
b. To find the maximum height reached by the ball, we need to calculate the vertical distance covered by the ball. Since there is a vertical acceleration due to gravity, we can use the equation for vertical displacement:
Vertical displacement = (vertical velocity² × sin²(angle)) / (2 × acceleration due to gravity) (where angle is the launch angle in radians)
The maximum height is the vertical displacement from the ground level.
c. To find the time spent in the air, we can use the equation for the time of flight mentioned above.
Now let's calculate the answers:
Step 1: Convert the angle from degrees to radians.
angle = 70.0° × (π / 180)
Step 2: Calculate the horizontal distance.
time = (2 × vertical velocity) / acceleration due to gravity
Horizontal distance = horizontal velocity × time
Step 3: Calculate the maximum height.
Vertical displacement = (vertical velocity² × sin²(angle)) / (2 × acceleration due to gravity)
Maximum height = Vertical displacement
Step 4: Calculate the time of flight.
Time of flight = (2 × vertical velocity) / acceleration due to gravity
By following these steps and plugging in the given values (horizontal velocity = 55.0 m/s, angle = 70.0°, and acceleration due to gravity = 9.8 m/s²), you can determine the answers for each question.