⦁ A ball thrown horizontally at 25 m/s travels a horizontal distance of 50 m before hitting the ground. From what height was the ball thrown?
Xo = 25 m/s.
d = 50 m
Tf = Fall time.
d = Xo*Tf = 25 * Tf = 50 m
Tf = 2 s.
h = 0.5g*Tf^2 = 4.9*2^2 = 19.6 m.
Well, if the ball was thrown horizontally, then it must have been thrown from the ground, because if it was thrown from a height, that would be called throwing it at an angle. So, the height from which the ball was thrown is 0 meters.
To find the height from which the ball was thrown, we can use the kinematic equation:
d = v₁t + (1/2)at²
In this case, the ball is thrown horizontally, which means the initial vertical velocity (v₁y) is 0. The acceleration due to gravity (a) is -9.8 m/s² (negative because it acts downwards), and the distance traveled vertically (d) is what we need to find.
Given that the horizontal distance (dₓ) is 50 m, and the initial horizontal velocity (v₁x) is 25 m/s, we can calculate the time (t) it takes for the ball to travel that distance horizontally using the equation:
dₓ = v₁ₓt
Substituting the given values:
50 m = 25 m/s * t
Now, solve for t:
t = 50 m / 25 m/s = 2 s
Since the ball was thrown horizontally, the time it takes to fall vertically (t) is also 2 s.
Now, plug in the values for t and a into the first equation to solve for d:
d = 0 * 2 s + (1/2) * (-9.8 m/s²) * (2 s)²
Simplifying:
d = 0 + (-9.8 m/s²) * 2 s² = -19.6 m/s² * s² = -19.6 m
The negative sign indicates that the ball falls downward, so we need to take the magnitude of the distance. Thus, the height from which the ball was thrown is:
|d| = |-19.6 m| = 19.6 m
So, the ball was thrown from a height of 19.6 meters.
To find the height from which the ball was thrown, we can use the equations of motion for projectile motion. In this case, since the ball is thrown horizontally, the initial vertical velocity is zero and the only force acting on the ball is gravity.
The equation to calculate the height from which the ball was thrown is:
h = (1/2) * g * t^2
Where:
h is the height
g is the acceleration due to gravity (approximately 9.8 m/s^2)
t is the time it takes for the ball to hit the ground
To find t, we can use the horizontal distance traveled by the ball and its horizontal velocity.
t = d / v
Where:
d is the horizontal distance traveled (50 m)
v is the horizontal velocity (25 m/s)
Now, we can substitute this value of t into the equation to find the height:
h = (1/2) * g * (d / v)^2
Now, let's calculate the height:
h = (1/2) * 9.8 * (50 / 25)^2
h = 4.9 * (2)^2
h = 4.9 * 4
h = 19.6 meters
Therefore, the ball was thrown from a height of 19.6 meters.