A baseball is hit off a 1 meter high tee at an angle of 25 degrees and at a speed of 43 m/s. How far to the nearest tenth of a meter does it travel horizontally?
vertical problem
Vi = 43 sin 25 = 18.17
v = Vi - 9.8 t
0 = 18.17 - 9.8 t at top
t to top = 1.85 seconds to top
h = 1 + 18.17 (1.85) - 4.9 (1.85)^2
h = 1 + 33.61 - 16.77 = 17.8 m at top at 1.85 s
now fall from 17.8 m
17.8 = 4.9 t^2
t = 1.91 s to fall
total time in air = 1.85+1.91 = 3.76 s in the air
how far, horizontal problem
u = constant horizontal speed = 43 cos 25 = 39 m/s
distance = 39 * 3.76 = 147 meters
a football field and more than a half
To find the horizontal distance traveled by the baseball, we can use the basic principles of projectile motion.
The horizontal component of the initial velocity (Vx) will remain constant throughout the entire motion, as there are no horizontal forces acting on the ball.
Step 1: Find the horizontal component of the initial velocity.
Given that the initial velocity of the baseball is 43 m/s, we can determine the horizontal component by using the trigonometric function cosine. The horizontal component of the velocity, Vx, is given by:
Vx = V * cos(θ)
where V is the initial velocity and θ is the angle of projection.
Plugging in the values:
Vx = 43 m/s * cos(25°)
Step 2: Calculate the time of flight.
To calculate the horizontal distance traveled, we need to know the time of flight. The time of flight can be determined using the vertical component of the initial velocity and the acceleration due to gravity. Since the ball starts and ends at the same height, we can use the following equation:
Δy = Vy * t + (1/2) * g * t^2
where Δy is the change in height (in this case, 0), Vy is the vertical component of the initial velocity, t is the time of flight, and g is the acceleration due to gravity (approximately 9.8 m/s^2).
Plugging in the values:
0 = Vy * t + (1/2) * g * t^2
0 = V * sin(θ) * t + (1/2) * g * t^2
Step 3: Solve for the time of flight.
Rearranging the equation:
(1/2) * g * t^2 = - V * sin(θ) * t
This is a quadratic equation with t as the unknown. Solving the equation, we get two possible values for t. Since we are interested in the time of flight, we choose the positive value:
t = (-V * sin(θ) + √(V^2 * sin^2(θ) + 2 * (1/2) * g * 0)) / g
Plugging in the values:
t = (-43 m/s * sin(25°) + √(43 m/s)^2 * sin^2(25°) + 2 * (1/2) * (9.8 m/s^2) * 0) / (9.8 m/s^2)
Step 4: Calculate the horizontal distance traveled.
The horizontal distance, D, can be calculated as the product of the horizontal component of the initial velocity and the time of flight:
D = Vx * t
Plugging in the values:
D = (43 m/s * cos(25°)) * t
Now, you can compute the final answer to find the horizontal distance traveled by the baseball.