A roller coaster ride at an amusement park lifts a car of mass $ 700 \, \text{kg} $ to point $ A $ at a height of $ 90 \, \text{m} $ above the lowest point on the track, as shown above. The car starts from rest at $ A $, rolls with negligible friction down the incline and follows the track around a loop of radius $ 20 \, \text{m} $. Point $ B $, the highest point on the loop, is at a height of $ 50 \, \text{m} $ above the lowest point on the track.

Question

A roller coaster ride at an amusement park lifts a car of mass $ 700 \, \text{kg} $ to point $ A $ at a height of $ 90 \, \text{m} $ above the lowest point on the track, as shown above. The car starts from rest at $ A $, rolls with negligible friction down the incline and follows the track around a loop of radius $ 20 \, \text{m} $. Point $ B $, the highest point on the loop, is at a height of $ 50 \, \text{m} $ above the lowest point on the track.(a) Calculate the maximum speed of the car, when it reaches the very bottom of the track .
, (b) Calculate the speed $ v_B $.
, (c) Draw and label the vectors to represent the forces acting on the car when it is upside down at point $ B $.
, (d) Calculate the magnitude of all the forces identified in (c). If the shape or size of the loop were modified, what changes (if any) would be needed to ensure that the roller coaster still has the same speed at the top of the loop as calculated in part (b)? Justify your answer.
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Rachel · Accepted Answer

$\boxed{v_{\text{max}} = 42 \, \text{m/s}}$
$\boxed{v_B = 28 \, \text{m/s}}$
Forces at Point B: Normal Force ($ F_N $): Points towards the center of the loop. Gravitational Force ($ F_g $): Points
downwards (also towards the center when car is at the top of the loop).$\boxed{F_N = 20580 \, \text{N}}$, $\boxed{F_g = 6860 \, \text{N}}$

If the loop’s shape or radius is modified, the drop height should be adjusted accordingly to ensure energy conservation and keep $ v_B $ unchanged. This is supported by the equation derived in part B: $v_B = \sqrt{2g (h_A &#8211; h_B)}$ which shows that $ v_B $ depends only on the height difference of the rollercoaster.

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