A pitched ball is hit by a batter at a 45° angle and just clears the outfield fence, 107 m away. If the fence is at the same height as the pitch, find the velocity of the ball when it left the bat. Ignore air resistance.
To find the velocity of the ball when it left the bat, we can analyze the motion of the ball projectile.
Let's break down the given information:
- The ball was hit at a 45° angle, which means it was launched with an initial angle of 45° with respect to the horizontal.
- The ball just cleared the outfield fence, which is located 107 m away from the point where it was hit.
- The height of the fence is the same as the pitch, so we can assume there is no change in vertical position.
We can solve this problem by considering the horizontal and vertical components separately.
1. Analyzing the horizontal motion:
In the absence of air resistance, the horizontal velocity of the ball remains constant throughout the flight. We can assume this initial horizontal velocity as Vx.
Therefore, the horizontal displacement (distance travelled horizontally) can be calculated using the equation:
Horizontal displacement = Horizontal velocity × Time
In this case, the horizontal displacement is given as 107 m, and the time can be calculated using the vertical motion of the ball.
2. Analyzing the vertical motion:
The vertical motion can be analyzed using the standard equations of motion under constant acceleration. The only acceleration acting on the ball in the vertical direction is due to gravity (g = 9.8 m/s^2).
We can find the time taken for the ball to reach its maximum height (at the highest point of its trajectory) using the equation:
Vertical velocity = Initial vertical velocity + (Acceleration × Time)
At the highest point of the trajectory, the vertical velocity becomes zero, and we can use this to find the time taken to reach the maximum height.
Next, we can find the total time of flight by doubling the time taken to reach the maximum height (as the time of ascent equals the time of descent).
Finally, using the total time of flight, we can substitute it back into the equation for horizontal displacement to find the value of the horizontal velocity.
Thus, by calculating the horizontal velocity, we can determine the velocity of the ball when it left the bat.
Please note that this solution assumes no air resistance and ignores other factors like the ball's spin, which can affect the path of the ball in practice.