A 375 g ball is thrown vertically upward with a speed of 3.0 m/s. How high did it go? How fast is it moving when it is 0.25 m high?
The ball will reach a maximum height of 4.5 m. When it is 0.25 m high, it will be moving at a speed of 2.2 m/s.
To find out how high the ball went and its speed at a specific height, we can use the equations of motion for projectile motion.
1. Find the maximum height (h) reached by the ball:
The initial velocity (u) is 3.0 m/s, and the acceleration due to gravity (g) is approximately 9.8 m/s² (assuming no air resistance). We can use the following formula to find the maximum height reached by the ball:
h = (u²) / (2g)
Substituting the values into the formula:
h = (3.0 m/s)² / (2 * 9.8 m/s²)
h ≈ 0.459 meters
Therefore, the ball reached a maximum height of approximately 0.459 meters.
2. Find the speed of the ball when it is 0.25 meters high:
To find the speed of the ball at a specific height, we can use the equation of motion:
v² = u² - 2gh
Given that the initial velocity (u) is 3.0 m/s, the acceleration due to gravity (g) is 9.8 m/s², and the height (h) is 0.25 meters, we can solve for v.
v² = (3.0 m/s)² - 2 * 9.8 m/s² * 0.25 m
v² = 9 m²/s² - 4.9 m²/s²
v² ≈ 4.1 m²/s²
Taking the square root of both sides:
v ≈ √4.1 m²/s²
v ≈ 2.03 m/s
Therefore, the ball is moving at approximately 2.03 m/s when it is 0.25 meters high.
To determine how high the ball went, we can use the equations of motion. The first equation is:
v^2 = u^2 + 2as
where:
v = final velocity
u = initial velocity
a = acceleration
s = displacement
In this case, the ball is thrown vertically upward, so the acceleration will be equal to the acceleration due to gravity, which is approximately -9.8 m/s^2. Since we want to determine the maximum height reached, the final velocity at the highest point will be zero (v = 0 m/s).
Using the given information:
u = 3.0 m/s
a = -9.8 m/s^2
v = 0 m/s
Plugging these values into the equation, we can solve for the displacement (s):
0^2 = 3.0^2 + 2(-9.8)s
0 = 9 + (-19.6)s
19.6s = 9
s = 9 / 19.6
s ≈ 0.459 m
Therefore, the ball reached a height of approximately 0.459 meters.
To find the speed of the ball when it is at a height of 0.25 m, we can use the second equation of motion:
v = u + at
Using the given information:
u = 3.0 m/s
a = -9.8 m/s^2
s = 0.25 m
Plugging these values into the equation, we can solve for the final velocity (v):
v = 3.0 + (-9.8)t
t = (v - u) / a
t = (0 - 3.0) / (-9.8)
t ≈ 0.306 s
Now, we can find the final velocity:
v = 3.0 + (-9.8)(0.306)
v ≈ 3.0 - 3.00
v ≈ 0.00 m/s
Therefore, the ball is moving at approximately 0.00 m/s when it is at a height of 0.25 m.