A target lies flat on the ground 4 m from the side of a building that is 10 m tall, as shown below.
The acceleration of gravity is 10 m/s2 . Air resistance is negligible.
A student rolls a 4 kg ball off the horizontal roof of the building in the direction of the target.
Look out below!!
Was there a question in there somewhere?
To determine the time it takes for the ball to reach the target, we can use the equations of motion and consider the vertical motion and horizontal motion separately.
First, let's analyze the vertical motion of the ball. We know the initial velocity is zero (as the ball is rolled), the final height is 4 m (from the ground to the target), and the acceleration due to gravity is 10 m/s^2. We can use the equation:
h = ut + (1/2)gt^2
where:
h is the vertical displacement (4 m),
u is the initial velocity (0 m/s),
g is the acceleration due to gravity (-10 m/s^2),
t is the time.
Plugging in the values, we can solve for t:
4 = 0 + (1/2)(-10)t^2
4 = -5t^2
t^2 = -4/5
t = sqrt(-0.8) (taking the positive square root because time cannot be negative)
However, we encounter an issue. The time we calculated involves taking the square root of a negative number, which is not possible in real numbers. This indicates that the ball will not reach the target vertically.
Therefore, we need to analyze the horizontal motion of the ball. We know the horizontal distance is 10 m (the length of the building), and the ball rolls off the roof with no initial vertical velocity.
To determine the time it takes for the ball to travel horizontally, we can use the equation:
s = ut + (1/2)at^2
where:
s is the horizontal distance (10 m),
u is the horizontal initial velocity (unknown),
a is the horizontal acceleration (0 m/s^2, as there is no horizontal acceleration),
t is the time.
The equation simplifies to:
s = ut
Plugging in the known values:
10 = u * t
We can now solve for u (the horizontal initial velocity):
u = 10 / t
Now, we need to find the value of t, the time it takes for the ball to reach the target horizontally. This value will be the same as the vertical time we calculated earlier (sqrt(-0.8)).
Plugging the value of t into the equation for u:
u = 10 / sqrt(-0.8)
Again, we encounter an issue. The value inside the square root is negative, which is not possible in real numbers. This indicates that the ball will not reach the target horizontally either.
In conclusion, based on the given information, it is not possible for the 4 kg ball to reach the target when rolled off the roof of the building.