A baseball is thrown at an angle of 20◦ relative to the ground at a speed of 25.8 m/s. The ball is caught 43.6152 m from the thrower.
The acceleration due to gravity is
9.81 m/s2 .
vertical problem
Vi = 25.8 sin 20 = 8.824 m/s
v = Vi - 9.81 t
at top v = 0
t = 8.824/9.81 = .900 seconds at top
(therefore 1.8 seconds in air by the way)
h = Vi t - (9.81/2) t^2
= 8.824(.9) - 4.9(.81)
= 7.94 - 3.97 = 3.97 meters high
horizontal problem
u = 25.8 cos 20 = 24.2
times time in air of 1.8 s
= 43.6 meters sure enough
To find the acceleration due to gravity, we need to calculate the vertical component of the ball's initial velocity and the time it takes for the ball to reach the catcher.
First, let's find the vertical component of the initial velocity (Vy):
Vy = V * sin(θ)
where V is the magnitude of the initial velocity (25.8 m/s) and θ is the angle of the throw relative to the ground (20°).
Vy = 25.8 m/s * sin(20°)
Vy ≈ 8.77 m/s
Next, let's calculate the time of flight (t) using the equation:
t = (2 * Vy) / g
where g is the acceleration due to gravity (9.81 m/s^2).
t = (2 * 8.77 m/s) / 9.81 m/s^2
t ≈ 1.78 s
Now, we will calculate the horizontal component of the initial velocity (Vx):
Vx = V * cos(θ)
Vx = 25.8 m/s * cos(20°)
Vx ≈ 24.41 m/s
Finally, we'll find the horizontal distance (d) traveled by the ball using the equation:
d = Vx * t
d = 24.41 m/s * 1.78 s
d ≈ 43.46 m
The ball is caught 43.46 m from the thrower.
Therefore, the acceleration due to gravity is approximately 9.81 m/s^2.
To find the acceleration due to gravity, we can use the formula for the horizontal range of a projectile:
R = (v^2 * sin(2θ)) / g,
where:
- R is the horizontal range (43.6152 m in this case)
- v is the initial velocity (25.8 m/s in this case)
- θ is the angle of projection (20◦ in this case)
- g is the acceleration due to gravity (the unknown)
Rearranging the formula to solve for g:
g = (v^2 * sin(2θ)) / R.
Let's substitute the given values into the equation:
g = (25.8^2 * sin(2 * 20◦)) / 43.6152.
Calculating the values:
g = (662.64 * sin(40◦)) / 43.6152.
Note: In the equation, the angle θ is doubled because the horizontal range formula uses the angle of projection twice.
Now, we calculate the value of sin(40◦) and substitute it back:
sin(40◦) ≈ 0.6428.
g = (662.64 * 0.6428) / 43.6152.
g ≈ 9.71 m/s^2.
Therefore, the acceleration due to gravity is approximately 9.71 m/s^2.