A pitched ball is hit by a batter at a 45 degree angle. It just clears the outfield fence, 97 m away. Find the velocity of the ball when it left the bat. Assume the fence is the same height as the pitch.
A pitched ball is hit by a batter at a 45 degress angle and just clears the outfield fence, 99m away. If the fence is at the height add the pitch, find the velocity of the ball when it left the bat
To find the velocity of the ball when it left the bat, we need to make use of the kinematic equations of motion. Specifically, we can use the equation for horizontal motion and the equation for vertical motion to solve for the initial velocity of the ball.
Let's break the motion into horizontal and vertical components. Considering the horizontal component, we can assume there is no acceleration acting on the ball. Therefore, the horizontal velocity remains constant throughout the projectile motion.
The equation for horizontal motion is:
horizontal displacement = horizontal velocity × time
In our case, the horizontal displacement is given as 97 m, and the time of flight is the same for both horizontal and vertical components since the ball just clears the fence. Hence, we can start by calculating the time of flight using the vertical component.
Now, let's consider the vertical component. We assume that the ball is launched at an angle of 45 degrees above the horizontal. The equation for vertical motion is:
vertical displacement = (initial vertical velocity × time) - (0.5 × acceleration × time^2)
In our case, the vertical displacement is equal to the height of the fence, the initial vertical velocity is what we have to calculate, time is the same for both horizontal and vertical components, and the acceleration due to gravity is approximately 9.8 m/s² (acting downwards).
Since the ball just clears the fence, the vertical displacement is equal to the height of the fence. We can assume that the height of the fence is the same as the pitch.
Now, let's solve these equations simultaneously to find the initial velocity of the ball:
1) vertical displacement = (initial vertical velocity × time) - (0.5 × acceleration × time^2)
height_fence = (initial vertical velocity × time) - (0.5 × 9.8 m/s² × time^2)
2) horizontal displacement = horizontal velocity × time
97 m = horizontal velocity × time
Since the time is the same in both equations, we can isolate the time in the second equation and substitute it into the first equation:
time = 97 m / horizontal velocity
Substituting this time value in the first equation:
height_fence = (initial vertical velocity × (97 m / horizontal velocity)) - (0.5 × 9.8 m/s² × (97 m / horizontal velocity)^2)
Simplifying:
height_fence = (initial vertical velocity × 97 m / horizontal velocity) - (0.5 × 9.8 m/s² × (97 m)^2 / horizontal velocity^2)
Now, we can solve this equation for the initial vertical velocity.
Finally, we can use the horizontal displacement and the horizontal time of flight to find the horizontal velocity.
horizontal velocity = horizontal displacement / time
Substituting the calculated time:
horizontal velocity = 97 m / time
Please provide the height of the fence to continue.