Question

In the figure, a solid 0.3 kg ball rolls smoothly from rest (starting at height H = 5.9 m) until it leaves the horizontal section at the end of the track, at height h = 1.9 m. How far horizontally from point A does the ball hit the floor?

Answer 1

The ball hits the floor approximately 5.50 meters horizontally from point A.

Step-by-step explanation:

To find the horizontal distance the ball travels before hitting the floor, we can use the principle of conservation of energy. The initial potential energy of the ball at height H is equal to the sum of its final potential energy at height h and its final kinetic energy at the moment of impact.

Using the formula for potential energy (PE = mgh) and kinetic energy (KE = 0.5mv^2), we can set up the equation:

mgh = mgh + 0.5mv^2

Simplifying and solving for v, we get:

v = sqrt(2g(h - h))

Substituting the given values, we have:

v = sqrt(2 * 9.8 * (5.9 - 1.9)) = sqrt(78.4) = 8.85 m/s

Now, we can use the formula for horizontal distance (d = vt) to find the distance the ball travels:

d = 8.85 m/s * t

where t is the time it takes for the ball to hit the floor.

Since the ball starts from rest, we can use the equation:

h = 0.5gt^2

Substituting the given values, we have:

1.9 m = 0.5 * 9.8 * t^2

Solving for t, we get:

t = sqrt(1.9 / (0.5 * 9.8)) = sqrt(0.3878) = 0.622 s

Finally, substituting the value of t into the formula for horizontal distance, we get:

d = 8.85 m/s * 0.622 s = 5.50 m

Answer 2

To find the horizontal distance a ball will travel from a height, first use conservation of energy to determine the ball's horizontal velocity at the end of the track. Then apply projectile motion equations to calculate the time of fall and multiply by the horizontal velocity to get the distance.

Step-by-step explanation:

To solve for the horizontal distance the ball will travel from point A to the floor, we can separate the problem into two parts: the first part involves finding the horizontal velocity with which the ball leaves the track, and the second part involves using this velocity to determine how far the ball travels horizontally while falling from height h to the floor.

Step 1: Horizontal Velocity Calculation

We can use conservation of energy to find the ball's velocity when it leaves the track at height h = 1.9 m. Initially, the ball has potential energy (PE) due to its height H = 5.9 m and no kinetic energy since it starts from rest. At the end of the track, it has lost some potential energy and gained kinetic energy but no energy is lost to friction since it rolls smoothly:

• Initial PE = m x g x H
• Final PE = m x g x h
• Initial KE = 0 (since initial velocity is 0)
• Final KE = 0.5 x m x v2

Conservation of energy states that Initial PE = Final PE + Final KE, which can be rearranged to solve for velocity (v): v = √(2 x g x (H - h)). Substituting H = 5.9 m, h = 1.9 m, and g = 9.8 m/s2, we find the velocity.

Step 2: Projectile Motion

Once the ball leaves the track with the calculated horizontal velocity, it undergoes projectile motion in free fall from the height of h = 1.9 m. We calculate the time to fall using the formula for vertical motion under gravity: h = 0.5 x g x t2, where t is the time to fall. Having found t, we will then calculate the horizontal distance (x) the ball travels using x = v x t.

Combining these steps gives us the horizontal distance the ball travels from point A to the point of impact on the floor.