1.Two billiard balls each of mass 50g moving in opposite direction with a speed of 36km/h collide and rebound with same velocity.What is the impulse imparted to each balls due to the other ball?

2.A ball of mass 100g falls from a height of 5m and rebounds to a height of 1.25m.Calculate the impulse and average force between the ball and the ground if the time of contact is 0.1sec.

3.A cricket ball of mass 2kg moving with a velocity of 20m/s is brought to rest by a player in 0.1 second.

a)Find the impulse of the ball

b)The force applied by the player

4.An aeroplane travelling at a speed of 500km/h bends its wings at an angle of 30 degree with the vertical when it takes a turn.What's the radius of the turn?

5.A motorcyclist goes along a circular path of radius 320m at a speed of 144km/h.What's the angle of bending?

Let's break down each of your questions step by step, ensuring we understand the concepts of impulse, momentum, and forces involved in these scenarios.

1. Impulse from the Collision of Billiard Balls

When two billiard balls collide and rebound with the same velocity, we can calculate the impulse imparted to each ball. Impulse is defined as the change in momentum of an object.

Calculating Impulse

First, we need to convert the mass of the balls from grams to kilograms:

• Mass of each ball, m = 50 g = 0.05 kg

The initial velocities of the balls are:

• Ball 1: v1 = 36 km/h = 10 m/s (since 36 km/h = 36/3.6 m/s)
• Ball 2: v2 = -36 km/h = -10 m/s (opposite direction)

After the collision, both balls rebound with the same speed but in opposite directions:

• Ball 1: v1' = -10 m/s
• Ball 2: v2' = 10 m/s

The change in momentum (impulse) for each ball can be calculated as:

• Impulse (J) = m(v' - v)

For Ball 1:

• J1 = 0.05 kg (-10 m/s - 10 m/s) = 0.05 kg * (-20 m/s) = -1 kg·m/s

For Ball 2:

• J2 = 0.05 kg (10 m/s - (-10 m/s)) = 0.05 kg * (20 m/s) = 1 kg·m/s

Thus, the impulse imparted to each ball is 1 kg·m/s in magnitude, but in opposite directions.

2. Impulse and Average Force of a Falling Ball

Next, let's analyze the ball that falls from a height and rebounds. We need to calculate the impulse and the average force exerted during the contact with the ground.

Finding Impulse

First, we calculate the velocity just before hitting the ground using the formula:

• v = √(2gh), where g = 9.81 m/s² and h = 5 m.

Calculating the velocity:

• v = √(2 * 9.81 m/s² * 5 m) = √(98.1) ≈ 9.9 m/s

Now, the ball rebounds to a height of 1.25 m, so we find the rebound velocity:

• v' = √(2 * 9.81 m/s² * 1.25 m) = √(24.525) ≈ 4.95 m/s

The impulse can be calculated as:

• Impulse (J) = m(v' - v)

Substituting the values:

• m = 0.1 kg, J = 0.1 kg (4.95 m/s - (-9.9 m/s)) = 0.1 kg * (4.95 + 9.9) = 0.1 kg * 14.85 m/s = 1.485 kg·m/s

Calculating Average Force

The average force can be calculated using the formula:

• Average Force (F) = Impulse / Time

Substituting the values:

• F = 1.485 kg·m/s / 0.1 s = 14.85 N

3. Impulse and Force on a Cricket Ball

Now, let's consider the cricket ball scenario. We need to find the impulse and the force applied by the player.

Impulse Calculation

The initial momentum of the ball is:

• p_initial = m * v = 2 kg * 20 m/s = 40 kg·m/s

Since the ball is brought to rest, the final momentum is 0. Thus, the impulse is:

• Impulse (J) = p_final - p_initial = 0 - 40 kg·m/s = -40 kg·m/s

Force Calculation

Using the impulse to find the average force:

• F = J / t = -40 kg·m/s / 0.1 s = -400 N

The negative sign indicates the force is applied in the opposite direction of the ball's motion.

4. Radius of Turn for an Aeroplane

For the aeroplane turning at an angle, we can use the following relationship:

Understanding the Forces

The lift force can be resolved into two components: vertical and horizontal. The horizontal component provides the centripetal force required for the turn.

• Lift = Weight (mg) = L = m * g
• Horizontal component of lift = L * sin(θ)

Setting the horizontal component equal to the centripetal force:

• L * sin(θ) = (m * v²) / r

Rearranging gives us:

• r = (m * v²) / (L * sin(θ))

Substituting the values (assuming L = mg):

• r = (v²) / (g * sin(θ))

Converting speed to m/s:

• 500 km/h = 500/3.6 ≈ 138.89 m/s

Now substituting values:

• g = 9.81 m/s², θ = 30 degrees (sin(30) = 0.