a pitcher throws a softball horizontally from a height of 1.25 m and notes that it strikes the ground a distance of 15 m from the point where it was thrown. how long did it take the ball to reach the ground? what was the velocity at which the ball was thrown? what was the vertical component of the ball's velocity when it hit the ground?
Show steps please!
To find the time it took for the ball to reach the ground, we can use the kinematic equation for horizontal motion:
distance = velocity × time
In this case, the distance is 15 m, and since the ball was thrown horizontally, the vertical distance it traveled is 0. Therefore, we can write:
15 m = velocity × time
Now let's find the velocity at which the ball was thrown. Since the ball was thrown horizontally, there is no vertical acceleration and the vertical velocity is constant. The initial vertical velocity is zero because the ball was thrown horizontally. The vertical component of the velocity remains zero throughout the motion.
So, the velocity at which the ball was thrown is the horizontal component of the velocity. Let's denote it as Vx.
To find Vx, we can use the kinematic equation for horizontal motion:
distance = velocity × time
Since the vertical distance is 0, the equation becomes:
0 m = Vx × time
As time is the same in both equations, we can equate the two equations:
15 m = Vx × time
Therefore, the velocity at which the ball was thrown, Vx, is 15 m divided by the time.
To find the vertical component of the ball's velocity when it hits the ground, we can use the equation for vertical motion:
final velocity = initial velocity + acceleration × time
Since the ball falls vertically downwards, the acceleration is 9.8 m/s² (acceleration due to gravity). The final velocity is zero since the ball hits the ground. The initial velocity in the vertical direction is zero because the ball was thrown horizontally. Let's denote the vertical component of velocity as Vy.
So the equation becomes:
0 m/s = 0 m/s + (9.8 m/s²) × time
Now, we have two equations involving time:
1) 15 m = Vx × time
2) 0 m/s = (9.8 m/s²) × time
Let's solve for time using equation 2:
0 m/s = (9.8 m/s²) × time
0 = 9.8 × time
time = 0 s
This means the ball took 0 seconds to reach the ground vertically. Since the horizontal and vertical motions are independent, the time for the horizontal motion is the same.
If we substitute this value of time into equation 1:
15 m = Vx × 0 s
Vx = undefined
Since the time is 0 s, the horizontal velocity, Vx, is undefined. This is because the ball does not undergo any horizontal acceleration, so it would require an infinite velocity to cover a distance of 15 m in 0 time.
To summarize:
- The ball took 0 seconds to reach the ground vertically.
- The velocity at which the ball was thrown horizontally, Vx, is undefined.
- The vertical component of the ball's velocity when it hit the ground, Vy, is 0 m/s.