A baseball is hit from home plate and it winds up in the stands as a homerun. From the point it was hit, it lands a horizontal displacement of 121.9 meters and a vertical displacement of 21.2 meters (in +y direction). If it took 5 seconds to travel from home plate to the stands, what would the range of the baseball been if the stands were not there, in meters. In other words, if the stands weren't there what is the range for the vertical displacement to be zero meters?
well, the x speed (v cosθ) is clearly 121.9/5 m/s
Now you can get the y-speed and θ, and work out the range.
To find the range of the baseball if the stands were not there, we need to calculate the horizontal distance traveled by the ball.
We can use the horizontal displacement and the time taken to find the horizontal velocity of the baseball. The horizontal velocity remains constant unless affected by external factors like air resistance.
The formula for horizontal displacement is given by:
Horizontal Displacement = Horizontal Velocity x Time
Since the horizontal displacement is given as 121.9 meters, and the time taken is 5 seconds, we can rearrange the formula to solve for the horizontal velocity:
Horizontal Velocity = Horizontal Displacement / Time
Horizontal Velocity = 121.9 meters / 5 seconds
Horizontal Velocity = 24.38 meters per second
Now, to find the range, we need to consider the time taken for the vertical displacement to return to zero. Since the vertical displacement is 21.2 meters and the acceleration due to gravity pulls the ball downward, we can use the equation for vertical displacement under constant acceleration:
Vertical Displacement = Initial Vertical Velocity × Time + (1/2) × Acceleration × Time^2
The initial vertical velocity is zero since it starts from rest when hit, and the acceleration due to gravity is approximately -9.8 m/s^2 (negative due to downward motion).
The equation simplifies to:
Vertical Displacement = (1/2) × Acceleration × Time^2
Rearranging the equation, we can solve for time:
Time = sqrt(2 × Vertical Displacement / Acceleration)
Time = sqrt(2 × 21.2 meters / 9.8 m/s^2)
Time ≈ 2.04 seconds
So, if the stands were not there, it would take approximately 2.04 seconds for the vertical displacement to return to zero.
Finally, to find the range, we multiply the horizontal velocity by the time taken for the vertical displacement to return to zero:
Range = Horizontal Velocity × Time
Range = 24.38 meters per second × 2.04 seconds
Range ≈ 49.8 meters
Therefore, if the stands were not there, the range of the baseball would be approximately 49.8 meters.