4- A single frictionless roller-coaster car of mass m = 750 kg tops the first hill with speed v= 15 m/s at height h = 40 m as shown i- Find the speed of the car at B and C First hill- h/2 ii- If mass m were doubled, would the speed at B increase, decrease, or remain the same?
Gravitational force
In nature, every object is attracted by every other object. This phenomenon is called gravity. The force associated with gravity is called gravitational force. The gravitational force is the weakest force that exists in nature. The gravitational force is always attractive.
Acceleration Due to Gravity
In fundamental physics, gravity or gravitational force is the universal attractive force acting between all the matters that exist or exhibit. It is the weakest known force. Therefore no internal changes in an object occurs due to this force. On the other hand, it has control over the trajectories of bodies in the solar system and in the universe due to its vast scope and universal action. The free fall of objects on Earth and the motions of celestial bodies, according to Newton, are both determined by the same force. It was Newton who put forward that the moon is held by a strong attractive force exerted by the Earth which makes it revolve in a straight line. He was sure that this force is similar to the downward force which Earth exerts on all the objects on it.
![**Problem Description:**
A single frictionless roller-coaster car of mass \( m = 750 \, \text{kg} \) tops the first hill with speed \( v = 15 \, \text{m/s} \) at a height \( h = 40 \, \text{m} \) as shown in the diagram.
**Tasks:**
i. Find the speed of the car at points B and C.
ii. If mass \( m \) were doubled, would the speed at B increase, decrease, or remain the same?
**Diagram Explanation:**
The diagram shows a roller-coaster track with the car at the top of the first hill at height \( h = 40 \, \text{m} \). Points B and C are at lower heights on the track with values not specified in the text. The graph emphasizes the roller-coaster's peaks and valleys, crucial for visualizing changes in height and related speeds.
**Solution Approach:**
**Part i:**
To find the speed of the car at points B and C, we can use energy conservation principles. Since the track is frictionless, the mechanical energy (sum of kinetic and potential energy) of the car will be conserved.
- At the top of the first hill:
\[
E_{\text{initial}} = K_{\text{initial}} + U_{\text{initial}} = \frac{1}{2}mv^2 + mgh
\]
Substituting the given values \( m = 750 \, \text{kg} \), \( v = 15 \, \text{m/s} \), \( h = 40 \, \text{m} \):
\[
E_{\text{initial}} = \frac{1}{2} \times 750 \, \text{kg} \times (15 \, \text{m/s})^2 + 750 \, \text{kg} \times 9.8 \, \text{m/s}^2 \times 40 \, \text{m}
\]
\[
E_{\text{initial}} = 84375 \, \text{J} + 294000 \, \text{J}
\]
\[
E_{\text{initial}} = 378375 \, \text{J}
\]
- At point B:](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0f7775f3-edad-4f27-ba1b-f7886a7505bd%2Fc3f449d0-722a-443f-802a-8dc374a55398%2Fcrtaz2_processed.jpeg&w=3840&q=75)
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