The question has three parts, it looks like a lot but it is just description, trying to understand relationships.
1. You toss a coin straight up into the air. Sketch on the axes below the velocity-time acceleration-time graphs of the coin from the instant it leaves your hand until the instant it returns to your hand. Assume that the positive direction y is upward. Indicate with arrows on your graphs the moment when the coin reaches its highest point.
2. Based on your knowledge of the gravitational force near the surface of the Earth, and on Newton’s second law, explain the sign and magnitude of the acceleration of the coin in question 1 during the three portions of its motion: on the way up, at the instant when it reaches its highest point, and on the way down.
3. Based on the way the velocity is changing, explain the sign and magnitude of the acceleration of the coin in question 1 during the three portions of its motion: on the way up, at the instant when it reaches its highest point, and on the way down.
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1. To sketch the velocity-time and acceleration-time graphs of the coin, we need to understand the motion of the coin while it is in the air.
When the coin leaves your hand, it starts moving upwards against the force of gravity. Its velocity initially increases in the positive direction (upwards), as it gains height. The velocity-time graph during this phase will show a positive slope, indicating the increasing velocity. The arrow indicating the moment when the coin reaches its highest point will be at the peak of this upward slope.
At the highest point, the velocity of the coin momentarily becomes zero. This is because the coin reaches its maximum height and starts slowing down due to the gravitational force acting on it. The velocity-time graph at this instant will show a horizontal line, indicating a velocity of zero.
As the coin descends back towards your hand, its velocity starts increasing again, but in the negative direction (downwards). The velocity-time graph during this phase will show a negative slope, indicating the decreasing velocity. The arrow indicating the moment when the coin returns to your hand will be at the lowest point of this downward slope.
The acceleration-time graph will help us understand the acceleration experienced by the coin during each phase of its motion. The acceleration due to gravity near the surface of the Earth is always directed downwards, regardless of the motion of the object. Therefore, throughout the entire motion of the coin, its acceleration will be directed downwards (negative y-direction). The magnitude of the acceleration will be constant since the gravitational force is constant near the surface of the Earth.
2. On the way up: During this phase, the coin is moving against the force of gravity. According to Newton's second law (F = ma), the net force acting on the coin is equal to its mass multiplied by the acceleration. Since the net force is directed opposite to the motion (upwards), the acceleration will also be directed opposite to the motion (upwards). Therefore, the sign of the acceleration will be negative (in the -y direction) during the coin's ascent. The magnitude of the acceleration will be equal to the gravitational acceleration near the surface of the Earth, which is approximately 9.8 m/s^2.
At the instant when it reaches its highest point: At this instant, the velocity of the coin momentarily becomes zero, meaning its motion is momentarily at rest. According to Newton's second law, an object at rest experiences a net force of zero. Therefore, there is no net force acting on the coin at this point, and hence, the acceleration is zero.
On the way down: During the descent, the force of gravity is acting in the same direction as the motion (downwards). Therefore, the acceleration will also be in the same direction as the motion (downwards). The sign of the acceleration will be positive (in the +y direction) during the coin's descent. The magnitude of the acceleration will still be equal to the gravitational acceleration near the surface of the Earth (approximately 9.8 m/s^2).
3. The sign and magnitude of the acceleration can be determined based on how the velocity is changing.
On the way up: As the coin moves upward, its velocity is decreasing since it is moving against the force of gravity. According to the relationship between velocity and acceleration (a = Δv/Δt), a negative change in velocity (Δv) divided by a positive change in time (Δt) will give us a negative acceleration. Therefore, the sign of the acceleration will be negative (in the -y direction). The magnitude of the acceleration will still be equal to the gravitational acceleration near the surface of the Earth (approximately 9.8 m/s^2).
At the instant when it reaches its highest point: The velocity of the coin momentarily becomes zero. The change in velocity (Δv) at this point is zero since the velocity is not changing. Dividing zero by any non-zero change in time (Δt) will give us zero acceleration. Therefore, the acceleration at this instant is zero.
On the way down: As the coin moves downward, its velocity is increasing since it is now moving in the same direction as the force of gravity. According to the relationship between velocity and acceleration (a = Δv/Δt), a positive change in velocity (Δv) divided by a positive change in time (Δt) will give us a positive acceleration. Therefore, the sign of the acceleration will be positive (in the +y direction). The magnitude of the acceleration will still be equal to the gravitational acceleration near the surface of the Earth (approximately 9.8 m/s^2).