A ball is thrown horizontally from the top of a building 17.6 m high. The ball strikes the ground at a point 67.7 m from the base of the building. The acceleration of gravity is 9.8 m/s2. Find the time the ball is in motion
h = 17.6 m.
g = 9.8 m/s^2
h = 0.5g*t^2 = 17.6 m
Solve for t.
1.895214166 s
To find the time the ball is in motion, we can use the equation of motion for the vertical component of the ball's motion. Since the ball is thrown horizontally, the initial vertical velocity is 0 m/s. The height of the building is the initial height (17.6 m), and the final height is the height of the ground (0 m). The acceleration due to gravity is -9.8 m/s^2 (negative because it acts downwards).
Using the formula:
delta_y = v_initial_y * t + (1/2) * a_y * t^2
where
delta_y is the change in height (final height - initial height),
v_initial_y is the initial vertical velocity,
a_y is the acceleration in the vertical direction,
and t is the time,
we can rearrange the equation to solve for t:
0 = 0 * t + (1/2) * (-9.8) * t^2
0 = -4.9 * t^2
This quadratic equation can be solved for t:
t^2 = 0 / -4.9
t = sqrt(0)
t = 0
Since t = 0, the ball stays at the top of the building for an infinitesimally small amount of time before falling. Therefore, the time the ball is in motion is 0 seconds.
To find the time the ball is in motion, we can use the kinematic equation:
d = v*t + (1/2)*a*t^2
Where:
d is the distance travelled (67.7 m in this case)
v is the initial horizontal velocity (which is equal to the final horizontal velocity since there is no horizontal acceleration)
a is the horizontal acceleration (zero in this case, since there is no horizontal force acting on the ball)
t is the time in motion (what we want to find)
Since there is no horizontal acceleration, the horizontal velocity remains constant throughout the motion. Therefore, we can find the horizontal velocity using the equation:
v = d/t
Substituting the given values, we have:
v = 67.7 m / t
Now, we need to find the vertical time of flight. Since the ball was thrown horizontally, the vertical component of the initial velocity is zero. We can use the equation for vertical displacement:
d = (1/2)*g*t^2
Where:
d is the vertical displacement (17.6 m in this case)
g is the acceleration due to gravity (9.8 m/s^2)
t is the time of flight (what we want to find)
Substituting the given values, we have:
17.6 m = (1/2)*9.8 m/s^2 * t^2
Simplifying this equation, we get:
8.8 m = 4.9 m/s^2 * t^2
Dividing both sides by 4.9 m/s^2, we get:
t^2 = 8.8 m / 4.9 m/s^2
t^2 = 1.796
Taking the square root of both sides, we get:
t = √(1.796)
t ≈ 1.34 s
Now, we can substitute this value of t into the equation for horizontal velocity to find the time of motion:
v = 67.7 m / t
67.7 m = v * t
67.7 m = (67.7 m / t) * t
67.7 m = 67.7 m
This equation is true, indicating that we have found the correct time of motion.
Therefore, the time the ball is in motion is approximately 1.34 seconds.