a steel ball rolls off a table top 60 cm high. if the ball strikes the floor at a distance 50 cm from the base of the table, what was its velocity at the instant it left the table?
To find the velocity of the steel ball at the instant it left the table, we can use the principle of conservation of energy.
The potential energy of the ball at the top of the table is converted into kinetic energy as it falls and moves horizontally. The total mechanical energy of the ball remains constant throughout the motion.
The potential energy (PE) of an object of mass (m) at a certain height (h) is given by the formula:
PE = m * g * h
where g is the acceleration due to gravity, which is approximately 9.8 m/s² on Earth.
The kinetic energy (KE) of the object is given by the formula:
KE = (1/2) * m * v²
where v is the velocity of the object.
Since the ball only moves horizontally after falling off the table, we can ignore the vertical motion (height) of the ball and focus on the horizontal distance traveled.
Given that the height of the table (h) is 60 cm and the distance from the base of the table to the point of impact (d) is 50 cm, we can determine the initial vertical speed of the ball using the equation of motion for vertical free fall:
d = (1/2) * g * t²
where t is the time it takes for the ball to hit the ground.
At the instant the ball leaves the table, its vertical velocity (Vy) is 0 because it hasn't begun falling yet. Therefore, the time it takes for the ball to hit the ground would be the same as the time it takes to fall from the table height.
Using the equation d = (1/2) * g * t², we can solve for t:
60 cm = (1/2) * (9.8 m/s²) * t²
Simplifying the equation:
1.96 * t² = 0.6
t² = 0.6 / 1.96
t² = 0.306
t ≈ √0.306
t ≈ 0.553 seconds
We can now find the horizontal velocity (Vx) of the ball using the equation of motion for horizontal motion:
Vx = d / t
Since the horizontal distance (d) from the base of the table to the point of impact is given as 50 cm, we can convert it to meters by dividing by 100:
d = 50 cm / 100 = 0.5 m
Now, we can calculate Vx using the formula:
Vx = 0.5 m / 0.553 s ≈ 0.903 m/s
Therefore, the velocity of the steel ball at the instant it left the table was approximately 0.903 m/s.