## Steel ball

### Task number: 2226

We will release a steel ball from rest down a smooth angled plane upon which it moves with acceleration of 0,5 m·s^{−2}. Then it moves to a level plane. In total, it travels 20 meters in 12 seconds. For how long did it move on the angled plane?

Neglect friction of the environment and the plane.

#### List of known information

*a*= 0,5 m·s^{−2}ball’s acceleration *s*= 20 mtotal travelled distance *t*= 12 stotal time of balls movement *t*_{1}= ? (s)time of ball’s movement on the angled plane #### Hint 1: Distance and speed of ball on angled plane

Draw a picture and mark all needed quantities.

What distance will the ball travel on the angled plane and what speed will it reach?

#### Hint 2: Ball’s movement on level plane

What is the ball’s movement on the level plane?

What is its speed during this movement?

What distance does it travel?

#### Hint 3: Total distance of the ball

Express the total distance the ball has travelled.

Solve the quadratic equation that you get for time

*t*_{1}and figure out which root of the equation fits the task’s conditions.#### Overall Solution

Picture and marking of quantities:

The ball moves on the angled plane with a uniformly accelerated straightforward motion. Then, on the level plane, it moves with a uniform straightforward motion. On the angled plane, the ball moves for

\[s_1\,=\,\frac{1}{2}at_1^{2}\,.\]*t*_{1}with a set acceleration*a*. In this time it travels the distance of:And will reach the speed:

\[v\,=\,at_1\,.\]It will move on the level plane with this speed for time of

\[s_2\,=\,v\left(t-t_1\right)\,=\,at_1\left(t-t_1\right)\,.\]*t*-*t*_{1}and will travel the distance:Total distance will be:

\[s\,=\,\frac{1}{2}\,at_1^{2}\,+\,at_1\left(t-t_1\right)\,.\]We multiply the brackets:

\[s\,=\,-\frac{1}{2}\,at_1^{2}\,+\,at_1t\,.\]And after adjusting:

\[t_1^{2}\,-\,2tt_1\,+\,\frac{2s}{a}\,=\,0\,.\]By solving the quadratic equation, we get two roots (marked as

\[t_1\,=\,t\,-\,\sqrt{t^{2}\,-\,\frac{2s}{a}}\,.\]*t*_{1},*t*_{2}); only one fits:Because:

\[t_1\,<\,t\,.\]Numerically:

\[t_1\,=\,\left(12\,-\,\sqrt{(12)^{2}\,-\,\frac{2{\cdot} 20}{0{,}5}}\right)\,\mathrm{s}\,=\,4\,\mathrm{s}\,.\]#### Answer

The steel ball moves on the angled plane for:

\[t_1\,=\,t\,-\,\sqrt{t^{2}\,-\,\frac{2s}{a}}\,=\, 4\,\mathrm{s}\,.\]