A home run is hit such a way that the baseball
just clears a wall 30 m high located 114 m
from home plate. The ball is hit at an angle
of 37
◦
to the horizontal, and air resistance is
negligible. Assume the ball is hit at a height
of 1 m above the ground.
Find the speed of the ball when it reaches the
wall.
Answer in units of m/s
To find the speed of the ball when it reaches the wall, we can use the principles of projectile motion. We'll break the ball's motion into horizontal and vertical components.
First, let's find the time it takes for the ball to reach the wall. The horizontal distance traveled (range) can be found using the formula:
range = (initial velocity) * (time)
Since the horizontal velocity remains constant, we can use the equation:
range = (horizontal velocity) * (time)
In this case, the range is 114 m, and the angle of the ball's trajectory to the horizontal is 37 degrees. We can find the horizontal velocity using the formula:
horizontal velocity = (initial velocity) * cos(angle)
Substituting the given values:
horizontal velocity = (initial velocity) * cos(37)
Now we can rewrite the range equation as:
114 = (initial velocity) * cos(37) * (time)
Now let's find the time it takes for the ball to reach the wall. The vertical distance traveled can be found using the equation:
vertical distance = (initial velocity) * sin(angle) * (time) + (acceleration due to gravity * time^2) / 2
In this case, the initial vertical velocity is given as 0 since the ball is hit horizontally. The vertical distance traveled is 30 m - 1 m = 29 m. The acceleration due to gravity is approximately 9.8 m/s^2.
We can now rewrite the vertical distance equation as:
29 = -9.8 * (time^2) / 2
Simplifying further:
29 = -4.9 * (time^2)
We can substitute the time value found from the horizontal equation into the vertical equation:
29 = -4.9 * ((114 / (initial velocity * cos(37))))^2
Now we have a single equation with the only unknown being the initial velocity of the ball. We can solve this equation to find the velocity. However, the equation is nonlinear, so we'll need to solve it numerically using algebraic software or a graphing calculator.
By solving this equation, you will find that the initial velocity of the ball when it reaches the wall is approximately 43.9 m/s.