# AP Physics B - 2013 #4 (Energy and Momentum)

TLDRThis video script dissects a 2013 AP Physics B exam problem involving mechanics and projectile motion. It explains how to calculate the velocity of a ball after a block hits a bumper, determine the block's acceleration, and find the mass of the block using Newton's laws. The script also addresses the effect of increasing the ball's mass on its horizontal distance traveled upon hitting the floor, demonstrating how acceleration and velocity are influenced by mass changes.

###### Takeaways

- 📚 The problem is from the 2013 AP Physics B exam and involves mechanics, specifically the motion of a block and a ball.
- 🏗 A 3 kg cup with negligible mass is attached to a block of mass m, connected by a string to a pulley with another object hanging from it.
- 🚀 The system is released from rest, and the block accelerates to the right, hitting a bumper that launches the ball horizontally.
- 🎯 The ball travels through the air and lands 1.8 m away from the base of the cup, which is on a table 2.4 m above the floor.
- ⏱ The key to solving for the ball's velocity (Vx) is understanding the projectile motion, which involves both vertical and horizontal components.
- 📉 The horizontal acceleration of the block is zero since there are no forces acting in the horizontal direction.
- 📈 To find Vx, the time it takes for the ball to fall is calculated using the vertical displacement and acceleration due to gravity.
- 🔢 The average velocity in the horizontal direction (Vx) is found by dividing the horizontal distance by the time of fall.
- 📊 For part B, the acceleration of the block is determined using the kinematic equation relating final velocity, initial velocity, and acceleration.
- 🔧 In part C, the mass of the block is calculated by analyzing the forces acting on the system and using Newton's second law.
- 🔄 Part D explains that increasing the mass of the ball will decrease the horizontal distance it travels upon hitting the floor due to a decrease in the system's acceleration and the ball's launching velocity.

###### Q & A

### What is the primary concept being tested in this AP Physics B exam problem?

-The primary concept being tested is mechanics, specifically involving projectile motion and dynamics of a system with pulleys and forces.

### Why is analyzing the initial segment of the problem (the block accelerating over 0.95 m) not the best approach to solve for the speed of the ball?

-Analyzing the initial segment is not the best approach because there are too many unknowns, such as the mass of the block and the forces involved. Instead, it is more straightforward to analyze the projectile motion after the block hits the bumper.

### How do you determine the time it takes for the ball to fall 2.4 meters?

-The time it takes for the ball to fall is determined by using the equation for motion in the vertical dimension: Δy = ½gt². By solving for time, t = √(2Δy/g), and plugging in the values (2.4 m for distance and 10 m/s² for acceleration due to gravity), we get the time as approximately 0.69 seconds.

### What is the horizontal velocity of the ball just after it leaves the cup?

-The horizontal velocity vₓ of the ball is determined using the distance it travels horizontally (1.8 m) and the time it takes to fall (0.69 seconds). The horizontal velocity is vₓ = 1.8 m / 0.69 s = 2.6 m/s.

### How is the acceleration of the block on the table calculated?

-The acceleration is calculated using the kinematic equation v_f² = v_i² + 2aΔd. Since the initial velocity v_i is 0 and the final velocity v_f is 2.6 m/s over a distance Δd of 0.95 m, the acceleration a is a = v_f² / (2Δd) = 2.6² / (2 × 0.95) = 3.55 m/s².

### What forces are considered when determining the dynamics of the system involving the pulley and the masses?

-The primary forces considered are the gravitational force on the hanging mass (mg) and the tension in the string, which transmits this force to the block on the table, causing it to accelerate.

### How is the mass of the block (big M) calculated in part C?

-The mass of the block (big M) is calculated by equating the net force to the total mass times acceleration (F = ma). Given the gravitational force on the hanging mass and the acceleration of the system, the equation is Mg = (M + m)a, where m is the mass of the hanging object. Solving for M gives M = (mg/a) - m.

### What effect does increasing the mass of the ball have on the horizontal distance it travels after being launched?

-Increasing the mass of the ball decreases the horizontal distance it travels. This is because a larger mass results in a lower acceleration of the system due to the same gravitational force, leading to a lower launch velocity and thus a shorter horizontal distance.

### Why does the acceleration of the block and the entire system decrease when the mass of the ball increases?

-The acceleration decreases because the force due to gravity on the hanging mass remains constant while the total mass of the system increases. This results in a lower overall acceleration since a = F/mₜₒₜₐₗ.

### What are the key equations used in solving this problem?

-The key equations used include the kinematic equations for projectile motion and linear motion, such as Δy = ½gt², vₓ = Δx/t, v_f² = v_i² + 2aΔd, and F = ma. These equations help determine time, velocity, acceleration, and forces in the system.

###### Outlines

##### 📚 Physics Problem-Solving Strategy

The paragraph discusses a problem from the 2013 AP Physics B exam involving mechanics. The scenario involves a cup with a block attached, a pulley system, and a projectile motion of a ball. The focus is on finding the velocity of the ball after it hits a bumper and before it leaves the cup. The speaker suggests not to focus on the initial acceleration but to analyze the projectile motion, using the horizontal distance and time to calculate the horizontal velocity (Vx). The method involves calculating the time it takes for the ball to fall using the vertical distance and gravity, then using this time to find the horizontal velocity by dividing the horizontal distance by the time. The paragraph concludes with the solution to part A of the problem, which is worth three points.

##### 🔍 Analyzing Kinematics for Constant Acceleration

This paragraph continues the physics problem-solving process, focusing on part B, which is to determine the acceleration of the block as it moves across the table. The speaker emphasizes the importance of recognizing the initial and final velocities and the distance over which the acceleration occurs. Using the kinematic equation v2^2 = u2^2 + 2as, where u is the initial velocity, v is the final velocity, a is the acceleration, and s is the distance, the speaker calculates the acceleration. The correct approach is to square the final velocity and divide by twice the distance, leading to the acceleration value of 3.55 m/s^2. The paragraph also includes a minor calculation error that is corrected, emphasizing the importance of attention to detail in such calculations.

##### 🔧 Applying Dynamics to Find the Mass of the Block

In this paragraph, the speaker tackles part C of the problem, which is to find the mass of the block. The approach involves analyzing the dynamics of the system, considering the forces acting on the block and the acceleration calculated in the previous paragraph. The speaker explains that the net force on the block is equal to the weight of the smaller object pulling it down, and since there is no friction, this force is the only one affecting the horizontal motion. The key insight is that the entire system, including the block, the cup, and the ball, accelerates at the same rate due to the force exerted by the smaller object's weight. The speaker then sets up an equation to solve for the mass of the block (M), using the previously calculated acceleration and the weight of the smaller object. After correcting a calculation mistake, the speaker finds the mass of the block to be 4.24 kg.

##### 📉 Effect of Increased Mass on Horizontal Distance

The final paragraph, part D, explores the effect of increasing the mass of the ball on the horizontal distance it travels when it hits the floor. The speaker explains that while the mass of an object does not affect the rate of fall due to gravity, it does affect the final speed of the object because the same force (mg) is now acting on a greater mass, leading to a decrease in acceleration. As a result, the final velocity of the ball when it hits the bumper is reduced, which in turn decreases the horizontal distance it travels during its fall. The speaker assigns points to the explanation, emphasizing the importance of understanding how changes in mass can affect the dynamics of a system.

###### Mindmap

###### Keywords

##### 💡Mechanics

##### 💡Projectile Motion

##### 💡Acceleration

##### 💡Force

##### 💡Mass

##### 💡Velocity

##### 💡Kinematics

##### 💡Dynamics

##### 💡Gravitational Acceleration

##### 💡Average Velocity

##### 💡Friction

###### Highlights

The problem is primarily mechanics-based and involves a 3 kg cup of negligible mass attached to a block on a table, with a string connected to a pulley and another object hanging from it.

The problem involves releasing the mass from rest, causing it to accelerate to the right and hit a bumper, launching a ball horizontally.

The ball falls 2.4 m below the table and lands 1.8 m away from the base, leading to questions about the ball's speed and block's acceleration.

To solve for the speed of the ball after the block hits the bumper, we need to analyze the projectile motion.

The acceleration in the horizontal dimension (X) is zero for an object with no force acting on it.

We need to determine the time it takes for the ball to fall 2.4 m to calculate the horizontal velocity (VX).

The initial vertical velocity (VY) is zero, and the vertical acceleration is due to gravity (G).

Using the equation for vertical distance and acceleration, we find the time to fall is approximately 0.69 seconds.

The horizontal velocity (VX) is then calculated using the horizontal distance (1.8 m) and the time (0.69 seconds), resulting in a velocity of 2.6 m/s.

To determine the block's acceleration, we use the final velocity and distance traveled, assuming constant acceleration.

The block's acceleration is found to be 3.55 m/s² using kinematic equations.

For part C, we need to calculate the mass of the block by analyzing the dynamics of the system.

The net force on the block is equal to the mass of the little object times gravity, pulling the big block with a force to the right.

All objects in the system accelerate together, with the same acceleration determined by the force on the little object.

The mass of the block is calculated to be 4.24 kg by dividing the force by the acceleration and accounting for the mass of the other objects.

Increasing the mass of the ball decreases the horizontal distance it travels when it hits the floor because the system's acceleration decreases.

###### Transcripts

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