a baseball is thrown at an angle of 24 degrees relative to the ground at a speed of 23.4m/s. the ball is caught 41.4798m from the thrower. how long is it in the air?
To calculate the time it takes for the baseball to be in the air, we can use the kinematic equations of motion. Let's break down the given information:
Angle of projection (θ): 24 degrees
Initial speed (v₀): 23.4 m/s
Distance traveled (d): 41.4798 m
To solve for the time the baseball is in the air, we can use the range equation:
d = (v₀² * sin(2θ)) / g
where:
d is the horizontal distance traveled,
v₀ is the initial speed of the baseball,
θ is the angle of projection,
g is the acceleration due to gravity (approximately 9.8 m/s²).
Rearranging the equation for time (t):
t = d / (v₀ * cos(θ))
Now let's substitute the given values:
t = 41.4798 m / (23.4 m/s * cos(24°))
To calculate cos(24°), convert the angle to radians:
cos(24°) ≈ cos(0.4189 rad) ≈ 0.919
t ≈ 41.4798 m / (23.4 m/s * 0.919)
Using a calculator, we can find:
t ≈ 1.86 seconds
Therefore, the baseball is in the air for approximately 1.86 seconds.