Exercise - 2

How to think about this problem:

1. First, identify what the teacher has given you. Here, the force is 20 N and the acceleration is 1 m/s².

2. Next, identify what the teacher is asking you to find. The question asks for the mass of the ball.

3. Now ask yourself: Which physics concept connects force, mass, and acceleration? This is Newton's second law of motion.

4. Recall the main formula: F = ma.

5. Since mass is not directly isolated in the formula, rearrange it to get m = F/a.

6. Substitute the given values and calculate the answer: m = 20/1 = 20 kg.

The key idea to understand is: when you know the force and acceleration, Newton's second law allows you to find the mass. So whenever you see force, acceleration, and mass in the same problem, think of F = ma first.

Before moving on, check yourself: Do you understand what is given, what is being asked, and why Newton's second law is the correct concept? If yes, you have understood the basic idea of this problem.

How to think about this problem:

1. First, identify what is given. The force is given as a vector, F = (6î - 8ĵ + 10k̂) N, and the acceleration is 1 m/s².

2. Identify what the teacher is asking. We need to find the mass of the ball.

3. Identify the physics concept. Force, mass, and acceleration are connected by Newton's second law: F = ma.

4. Since the force is given in vector form, first find its magnitude using |F| = √(Fₓ² + Fᵧ² + F_z²).

5. Once the magnitude of force is known, use m = F/a to find the mass.

The key idea is: when force is given in î, ĵ, k̂ form and mass is asked, first calculate the magnitude of the force and then apply Newton's second law.

Self-check: Do you understand what is given, what is being asked, why we find the force magnitude, and why Newton's second law is used? If yes, you understand the approach to this problem.

How to think about this problem:

1. First, identify what is given. The mass is 2 kg, the force is 10 N, the ball starts from rest, and the force acts for 5 seconds.

2. Identify what the teacher is asking. We need to find the velocity after 5 seconds.

3. Identify the physics concept. A constant force produces a constant acceleration, so this problem involves Newton's second law and the equations of uniformly accelerated motion.

4. First use F = ma to find the acceleration from the given force and mass.

5. Once acceleration is known, use v = u + at because the acceleration is constant.

6. The phrase 'initially at rest' tells us that u = 0.

7. Substitute the values and calculate the final velocity.

The key idea is: when a constant force acts on a body of constant mass, the acceleration is constant. Find acceleration using F = ma, then use v = u + at to find the velocity.

Self-check: Do you understand what is given, what is being asked, why F = ma is used first, why the acceleration is constant, and why v = u + at can be used? If yes, you understand the approach to this problem.

How to think about this problem:

1. First, identify what is given. The mass is 2 kg, the ball is initially at rest, and the force changes with time: F = (6t²î + 4tĵ) N.

2. Identify what the teacher is asking. The question asks for the velocity of the ball at time t.

3. Identify the physics concept. Since force is given, first use Newton's second law to find acceleration. Then use the relation a = dv/dt to find velocity.

4. Notice that the force is not constant; it depends on time t. Therefore, the acceleration also depends on time, so we cannot simply use v = u + at with a constant acceleration.

5. First calculate acceleration using a = F/m.

6. Then integrate acceleration with respect to time to obtain velocity.

7. Finally, use the information that the ball starts from rest, v(0) = 0, to determine the integration constant.

The key idea is: when force varies with time, first find a(t) using F = ma, then integrate a(t) with respect to time to get v(t).

Self-check: Do you understand what is given, what is being asked, why F = ma is used first, why integration is needed, and how the initial condition is used? If yes, you understand the approach to this problem.

How to think about this problem:

1. First, identify what is given. The mass is 3 kg, the ball starts from rest, the force depends on time, and we need the velocity at t = 3 s.

2. Identify what the teacher is asking. We need the velocity vector at a particular time, t = 3 s.

3. Identify the physics concept. This is a variable-force problem. Since force changes with time, acceleration also changes with time.

4. Use Newton's second law, F = ma, to convert the given force into acceleration.

5. Since acceleration is a function of time, use a = dv/dt and integrate acceleration with respect to time to obtain velocity.

6. The ball starts from rest, so use v = 0 at t = 0 to determine the integration constant.

7. Finally, substitute t = 3 s to get the required velocity.

The key idea is: when force varies with time, do not directly use v = u + at because acceleration is not constant. Instead, use F = ma, then integrate acceleration with respect to time to find velocity.

Self-check: Do you understand what is given, what is being asked, why acceleration must be found first, why integration is required, and why the initial-rest condition is important? If yes, you understand the approach to this problem.

How to think about this problem:

1. First, identify what is given. The force produced by the rocket is 100 N and the rocket is moving with a velocity of 200 m/s.

2. Identify what the teacher is asking. We need to find the rate of fuel consumption, that is, how much fuel is being expelled every second.

3. Identify the physics concept. This is a rocket propulsion problem involving variable mass and conservation of momentum.

4. The important formula is F = (dm/dt)v. Here, dm/dt represents the mass of fuel expelled per second and v represents the velocity associated with that expelled mass.

5. What does this formula mean physically? A rocket produces thrust by throwing fuel backward at high speed. The fuel carries momentum backward, and the rocket receives an equal and opposite momentum change, causing it to move forward. The faster the fuel is expelled, or the more fuel expelled per second, the greater the thrust produced.

6. Therefore, the formula F = (dm/dt)v tells us that the thrust of a rocket depends on two things: how much fuel is expelled every second and how fast that fuel is expelled.

7. In the real world, this is what happens inside rocket engines. Burning fuel produces high-speed exhaust gases that are expelled through the rocket nozzle. The backward momentum of the exhaust produces a forward thrust on the rocket.

8. Now use the formula: dm/dt = F/v. Substitute the given values to find the rate of fuel consumption.

The key idea is: a rocket moves forward because it throws mass backward. The greater the momentum carried away by the exhaust every second, the greater the thrust on the rocket.

Real-world understanding: When you see a rocket launching, the huge stream of hot gases shooting downward is not just a by-product. Those gases are carrying momentum downward, and the rocket gains an equal and opposite momentum upward. This is Newton's third law and conservation of momentum in action.

Self-check: Do you understand what is given, what is being asked, which physics concept is involved, what the formula means physically, and how the formula connects to the real-world operation of a rocket? If yes, you have understood both the mathematical and physical meaning of the problem.

How to think about this problem:

1. First, identify what is given. The mass is 150 g, the acceleration is 20 m/s², and the force acts for 0.1 s.

2. Identify what the teacher is asking. We need to find the average force acting on the ball.

3. Identify the physics concept. This problem is based on Newton's second law, F = ma. Since the question asks for average force and gives acceleration, we can directly use F_avg = ma_avg.

4. Always check the units before using the formula. The mass is given in grams, so convert 150 g into 0.15 kg.

5. Substitute mass and acceleration into F_avg = ma_avg to obtain the average force.

What does the formula mean physically? Newton's second law tells us that force is responsible for changing an object's motion. For a given mass, greater acceleration requires greater force. For the same acceleration, a heavier object needs more force.

Real-world understanding: When a bat hits a cricket ball, a large force acts on the ball for a very short time. That force changes the ball's velocity. In a real collision, the force may change during the contact time, so we often describe its overall effect using average force.

The key idea is: average force tells us the effective force that produces the observed change in motion over the interaction. When the average acceleration is known, use F_avg = ma_avg.

Self-check: Do you understand what is given, what is being asked, why the mass must be converted to kilograms, and why Newton's second law gives the average force?

How to think about this problem:

1. First, identify what is given. The mass is 150 g, the acceleration is 20 m/s², and the force acts for a short time of 0.1 s.

2. Identify what the teacher is asking. We need to find the impulsive force, not the impulse.

3. Identify the physics concept. This is an impulsive-force problem, which is connected to Newton's second law and impulse-momentum.

4. Since acceleration is directly given, the quickest way to find the force is F = ma.

5. Remember to convert the mass from grams to kilograms before using the formula: 150 g = 0.15 kg.

6. The formula F = ma tells us physically that a force is needed to change the motion of an object. For the same acceleration, a heavier object requires a greater force.

7. What does 'impulsive' mean in the real world? When a bat hits a cricket ball, a hammer hits a nail, or a football is kicked, a relatively large force acts for a very short time. This short-duration force is called an impulsive force.

8. The time of 0.1 s becomes important when finding impulse: J = FΔt. Impulse tells us how much the object's momentum changes during the short collision or impact.

Real-world understanding: When you kick a ball, your foot applies a force to the ball for a short period. That force changes the ball's velocity. A larger force or a longer contact time produces a larger change in momentum.

The key idea is: impulsive force describes the force during a short-duration interaction, while impulse describes the total effect of that force over the time interval.

Self-check: Do you understand the difference between impulsive force and impulse, why the mass must be converted into kilograms, and why F = ma is sufficient to find the force in this question?

How to think about this problem:

1. First, identify what is given. The ball has mass 1 kg, is dropped from a height of 10 m, and rebounds to the same height. The value of g is 10 m/s².

2. Identify what the teacher is asking. We need to find the impulse imparted by the ground during the collision.

3. Identify the physics concept. This is an impulse-momentum problem. The most important idea is: Impulse = change in momentum.

4. Before finding impulse, we need the velocity immediately before and immediately after the collision. Use the equation of motion to find the speed just before hitting the ground.

5. Because the ball rebounds to the same height, its speed after rebounding is equal to its speed before hitting the ground. However, the direction changes, so the velocities have opposite signs.

6. Take upward as positive. Therefore, the velocity before collision is negative and the velocity after collision is positive.

7. Use J = Δp = m(v₂ - v₁). The change in momentum is large because the ball's velocity changes from downward to upward.

What does the formula mean physically? Impulse tells us how much an object's momentum changes during a force acting over a short time. A larger change in velocity means a larger change in momentum and therefore a larger impulse.

Real-world understanding: When a ball hits the ground, the ground exerts a large force on the ball for a very short time. That force stops the ball's downward motion and then pushes it upward. The total effect of this short-duration force is the impulse.

The key idea is: when an object reverses its direction during a collision, the change in momentum is greater than when it simply stops. That is why the impulse here is 20√2 N·s rather than 10√2 N·s.

Self-check: Do you understand why we need both the incoming and outgoing velocities, why their signs are opposite, why the same rebound height gives the same speed, and why impulse is equal to the change in momentum?

How to think about this problem:

1. First, identify what is given. The ball has mass 1 kg, it falls from 40 m, rebounds to 10 m, and g = 10 m/s².

2. Identify what the teacher is asking. We need to find the impulse given by the ground during the collision.

3. Identify the physics concept. This is an impulse-momentum problem. The main idea is: Impulse = change in momentum.

4. We are not directly given the velocities before and after the collision, so we must calculate them from the given heights using v² = u² + 2gh.

5. The ball is moving downward before hitting the ground and upward after rebounding. Therefore, the two velocities have opposite signs. Take upward as positive.

6. Find the velocity just before collision from the 40 m falling height, and find the velocity just after collision from the 10 m rebound height.

7. Then use J = m(v₂ - v₁) to find the change in momentum.

What does the formula mean physically? Impulse measures the total effect of a force acting over a short time. It tells us how much the object's momentum changes during the collision.

Real-world understanding: When the ball hits the ground, the ground first has to stop the ball's downward momentum and then give it upward momentum so that it can rebound. Because the ball changes direction, the change in momentum is the combination of the incoming and outgoing momenta.

Notice something important: the ball falls from 40 m but rebounds only to 10 m. Therefore, it comes toward the ground faster than it leaves the ground. This is why the incoming speed is 20√2 m/s while the outgoing speed is only 10√2 m/s.

The key idea is: for a collision, always think about the momentum just before and just after the collision. Impulse is the difference between these two momenta.

Self-check: Do you understand why two heights are used, why the velocities have opposite signs, why we use impulse = change in momentum, and why the rebound height determines the velocity after the collision?