A ball traveling horizontally leaves a 0.80 m high table and lands 1.1 meter away.
A. Predict the time required for the pool ball to hit the ground.
B. What was the initial velocity of the ball?
To solve this problem, we can use the equations of motion under constant acceleration. In this case, the only force acting on the ball is gravity, which causes it to accelerate vertically downward.
Step 1: Find the time taken to hit the ground (A)
To find the time required for the pool ball to hit the ground, we can use the equation:
h = ut + (1/2)gt^2
where:
- h is the vertical distance (0.80 m)
- u is the initial vertical velocity (unknown)
- g is the acceleration due to gravity (approximately 9.8 m/s^2)
- t is the time
Rearranging the equation, we get:
t^2 + (2u/g)t - (2h/g) = 0
Since it's a quadratic equation, we can solve it using the quadratic formula:
t = (-b ± √(b^2 - 4ac))/(2a)
where:
- a = 1
- b = 2u/g
- c = -2h/g
Plugging in the values, we have:
t = [-(2u/g) ± √((2u/g)^2 - 4(1)(-2h/g))]/(2(1))
Simplifying further, we get:
t = [-2u/g ± √(4u^2/g^2 + 8h/g)]/2
t = (-u/g ± √(u^2/g^2 + 2h/g))
Since time cannot be negative in this case, we'll take the positive value of t:
t = (-u/g + √(u^2/g^2 + 2h/g))
Substituting the given values:
0.80 = (1/2)u + √(u^2 + 15.3)
Now, to solve this equation, you can use various methods such as trial and error, substitution, or numerical techniques. Unfortunately, there is no simple algebraic solution for this equation. Therefore, you will need to use numerical methods or calculators to find the value of t.
Step 2: Find the initial velocity (B)
Once we have the time (t), we can use the following equation to find the initial velocity (u):
v = u + gt
where:
- v is the final velocity (0 m/s since the ball hits the ground)
- u is the initial vertical velocity (unknown)
- g is the acceleration due to gravity (approximately 9.8 m/s^2)
- t is the time (calculated in Step 1)
Rearranging the equation, we get:
u = -gt
Plugging in the values, we have:
u = -(9.8 m/s^2)(time calculated in Step 1)
Now, you can substitute the time calculated in Step 1 into the equation to find the initial velocity of the ball.