A boy throws a ball downward from the top of a 45 meter high bridge at -3 m/s. With what speed does the ball hit the cold hard pavement below?
To find the speed at which the ball hits the pavement, we need to analyze the motion of the ball.
We are given that the ball is thrown downward from a height of 45 meters with an initial velocity of -3 m/s. Assuming there is no air resistance, we can use the laws of physics to calculate the final velocity.
The ball undergoes free fall acceleration due to gravity, which is approximately 9.8 m/s².
Let's break down the problem step by step:
Step 1: Calculate the time taken by the ball to reach the ground.
We can use the equation for displacement in free fall:
s = ut + (1/2)at²
Here,
s = displacement = -45 m (negative because the ball is thrown downward)
u = initial velocity = -3 m/s
a = acceleration due to gravity = -9.8 m/s² (negative because it acts in the opposite direction of the initial velocity)
t = time taken (which we need to find)
Plugging in the values, we have:
-45 = (-3)t + (1/2)(-9.8)(t²)
-45 = -3t - 4.9t²
Step 2: Solve the quadratic equation for t.
Rearranging the equation, we have:
4.9t² + 3t - 45 = 0
We can solve this quadratic equation using various methods, such as factoring, completing the square, or the quadratic formula. Let's use the quadratic formula:
t = (-b ± √(b² - 4ac))/(2a)
Here,
a = 4.9
b = 3
c = -45
Plugging in these values, we have:
t = (-3 ± √(3² - 4(4.9)(-45)))/(2(4.9))
Simplifying this expression, we get:
t ≈ 3.44 s (approximately)
Step 3: Calculate the final velocity of the ball.
We can use the equation for final velocity in free fall:
v = u + at
Here,
u = initial velocity = -3 m/s
a = acceleration due to gravity = -9.8 m/s²
t = time taken = 3.44 s
Plugging in these values, we have:
v = -3 + (-9.8)(3.44)
v ≈ -36.38 m/s
Since speed is a scalar quantity and only represents magnitude, the negative sign does not matter. Therefore, the speed at which the ball hits the pavement is approximately 36.38 m/s.