A ball thrown horizontally at 27 m/s travels a horizontal distance of 51 m before hitting the ground. From what height was the ball thrown?
Dx = Xo*Tf = 51 m.
27*Tf = 51.
Tf = 51 / 27 = 1.89 s. = Fall time.
h = Yo*Tf + 0.5g*Tf^2.
h = 0 + 4.9*(1.89)^3 = 17.50 m.
To find the height from which the ball was thrown, we can use the equation for horizontal motion:
distance = velocity × time
In this case, the distance traveled horizontally is 51 m and the horizontal velocity is 27 m/s. Since there is no vertical motion, the time taken is the same as the time taken for the ball to hit the ground.
Now, we need to find the time taken for the ball to hit the ground. We know that the only force acting on the ball in the horizontal direction is due to gravity, and it does not affect the horizontal motion. Therefore, the time taken for the ball to hit the ground can be found using the vertical motion equation:
distance = (1/2) × acceleration × time^2
Since the ball is falling vertically, the acceleration is given by the acceleration due to gravity, which is approximately 9.8 m/s^2. The distance fallen vertically is the height from which the ball was thrown. Let's call it h.
Using the equation, we have:
h = (1/2) × 9.8 m/s^2 × time^2
Since we need to find the time taken for the ball to hit the ground, we can rearrange the equation to solve for time:
time = sqrt((2 × vertical distance) / gravity acceleration)
Plugging in the values, we get:
time = sqrt((2 × h) / 9.8)
Now, we can substitute this expression for time into the horizontal motion equation:
distance = velocity × time
Plugging in the values, we get:
51 m = 27 m/s × sqrt((2 × h) / 9.8)
Now, we can solve this equation for h.