Define (i) potential energy and (ii) kinetic energy - Leaving Cert Physics - Question 6 - 2015

Kinetic energy is the energy of an object in motion. It is defined as the work needed to accelerate an object from rest to its current velocity, measured in Joules (J). The formula for kinetic energy (KE) is given by:

$KE = \frac{1}{2} mv^2$

where m is the mass of the object and v is its velocity.

The principle of conservation of energy states that energy cannot be created or destroyed, only transformed from one form to another. In the context of a roller-coaster, gravitational potential energy at the highest point is converted to kinetic energy as the car descends.

The difference in height between point A and point B is:

$100 \text{ m} - 30 \text{ m} = 70 \text{ m}$

Thus, the height difference is 70 m.

Change in potential energy (PE) can be calculated using the formula:

$PE = mgh$

where m is mass (850 kg), g is gravitational acceleration (9.8 m/s²), and h is height difference (70 m).

Thus,

$PE = 850 \times 9.8 \times 70 = 424050 \text{ J}$.

Since we have established that energy is conserved, the kinetic energy (KE) at point B equals the change in potential energy:

$KE = PE = 424050 \text{ J}$.

Using the kinetic energy formula, we can solve for velocity:

$424050 = \frac{1}{2} \times 850 \times v^2$

Solving this gives:

$v^2 = \frac{424050 \times 2}{850} \implies v \approx 33.3 \text{ m/s}$.

To calculate deceleration, we use the formula:

$a = \frac{\Delta v}{t}$

However, since we do not have time, we can use the fact that initial velocity (vB) is 33.3 m/s and final velocity (vC) is 0 m/s. Assuming constant deceleration over a distance (d = 30 m):

Using:

$v^2 = u^2 + 2ad$

Setting v = 0, u = 33.3 m/s, and d = -30 m:

$0 = (33.3)^2 + 2a(-30)\implies a \approx -17.55 \text{ m/s}^2$.

Using Newton's second law, the average force (F) can be calculated as:

$F = ma$

where m is mass (850 kg) and a is deceleration (17.55 m/s²). Thus,

$F = 850 \times 17.55 \approx 14917.5 \text{ N}.$ The average force required to stop the car is approximately 14917.5 N.