A baseball is hit at a height of 1.2 m with an unknown initial velocity at 36 ° above the horizontal. It just clears a barrier of height 27.0658871949 m at a horizontal distance of 59.9481592832 m. Find:
a) the initial speed;
b) The time to reach the barrier;
c) the velocity at the barrier;
_____ i + _____ j m/s
To solve this problem, we can use the principles of projectile motion. Since we have the height and horizontal distance, we can find the initial speed, the time to reach the barrier, and the velocity at the barrier by using the following equations:
a) Finding the initial speed:
The initial vertical velocity (Viy) can be found using the equation:
Viy = Vo * sin(θ)
Where Viy is the vertical component of the initial velocity, Vo is the initial speed, and θ is the angle of projection.
Using the given information, we can substitute the known values into the equation:
Viy = Vo * sin(36°)
Next, we can use the equation for vertical displacement (Δy) to solve for the initial speed (Vo):
Δy = Viy * t + (1/2) * g * t^2
Where Δy is the vertical displacement, t is time, and g is the acceleration due to gravity (approximately 9.8 m/s^2).
Since the baseball just clears the barrier, the vertical displacement is the height of the barrier, given as 27.0658871949 m. Setting up the equation:
27.0658871949 = Viy * t + (1/2) * 9.8 * t^2
Now we have two equations:
Viy = Vo * sin(36°)
27.0658871949 = Viy * t + 4.9 * t^2
Substituting the value of Viy from the first equation into the second equation, we can solve for Vo.
b) Finding the time to reach the barrier:
Using the previously mentioned equation for vertical displacement (Δy), we can solve for time (t) when the baseball reaches the barrier:
Δy = Viy * t + (1/2) * g * t^2
Substitute the known values:
27.0658871949 = Viy * t + 4.9 * t^2
Now, we can use the value of Vo obtained from part a) in this equation to find the time (t).
c) Finding the velocity at the barrier:
The velocity at the barrier can be found by using the equations for horizontal (Vix) and vertical components (Viy) of velocity:
Vix = Vo * cos(θ)
Viy = Vo * sin(θ)
Since we know the horizontal distance and time, we can use these values to find the horizontal velocity (Vix):
Vix = Δx / t
Finally, we can find the velocity at the barrier by combining the horizontal and vertical components of velocity:
Velocity at the barrier = Vix * i + Viy * j
Following these steps, you should be able to find the answers for a), b), and c).