The potential energy-displacement graph of a 0.5 kg ball moving along a frictionless track is shown in the given figure. At O, the velocity of the ball is 0 ms⁻¹ and potential energy is 30 J. Calculate the velocity of the ball at P, Q and R.

Given:

Mass of ball, $m = 0.5 \text{ kg}$

At point O:

$$\text{Velocity} = 0 \text{ ms}^{-1}$$

Potential energy = 30 J

Therefore, total mechanical energy of the ball:

$$E = 30 \text{ J}$$

Since the track is frictionless, mechanical energy remains constant.

At point P:

From the graph, potential energy at P = 20 J

Kinetic energy at P:

$$K.E. = 30 - 20 = 10 \text{ J}$$

$$\text{Using, } K.E. = \frac{1}{2}mv^2$$

$$10 = \frac{1}{2} \times 0.5 \times v^2$$

$$10 = 0.25v^2$$

$$v^2 = 40$$

$$v = \sqrt{40}$$

$$v\thickapprox6.3\text{ ms}^{-1}$$

Therefore, velocity at P is 6.3 ms$^{-1}$

At point Q:

From the graph, potential energy at Q = 30 J

$$K.E. = 30 - 30 = 0$$

Hence, $v = 0 \text{ ms}^{-1}$

Therefore, velocity at Q is: $0 \text{ ms}^{-1}$

At point R:

From the graph, potential energy at R = 40 J

Since the total mechanical energy of the ball is only 30 J, the ball cannot reach point R.

Therefore, velocity at R cannot be calculated because the ball never reaches R.