Motion In A Straight Line

To solve this problem, we can use the equations of motion. We are given that a car is moving with uniform accelerationand passes through two points, P and Q, with velocities of 30 km/h and 40 km/h, respectively. We want to find the velocity of the car at the midpoint between P and Q.

Let's break down the problem step by step:

1. Given data:
- Initial velocity at point P (u) = 30 km/h
- Final velocityat point Q (v) = 40 km/h
- Distance between P and Q (s)

2. We can use the following equation of motion to relate velocity, initial velocity, acceleration, and distance:

3. We want to find the velocity of the car (V) at the midpoint between P and Q. Let's call this point M.

4. First, let's find the acceleration (a) of the car using the data for points P and Q:

Solving for 'a':

5. Now, we can find the velocity at point M using the same equation of motion:

Taking the square root of both sides:

So, the velocity of the car at the midpoint between points P and Q is approximately 35.35 km/h.

To solve this problem, let's break it down step-by-step.

1. Understanding the Motion of the Car:

• The car accelerates from rest with a constant acceleration .
• We need to find the velocity of the car (and hence the horizontal velocity of the ball at the instant it is dropped) at .

The formula for velocity under constant acceleration is given by:

Here, (since the car starts from rest), , and . Substituting these values, we get:

This means at the moment the ball is dropped, it has a horizontal velocity of .

2. Understanding the Motion of the Ball After Being Dropped:

• The ball is subject only to gravitational acceleration in the vertical direction after being dropped.
• The horizontal velocity of the ball remains constant at since there are no forces acting in the horizontal direction.
• The vertical acceleration of the ball is downward.

3. Calculating the Vertical Velocity of the Ball at :

• Since the ball is dropped at , we are interested in its motion between and .
• The time of vertical motion .

The vertical velocity after a time is given by:

4. Resultant Velocity of the Ball at :

To find the magnitude of the resultant velocity, we use the Pythagorean theorem:

5. Acceleration of the Ball at :

The acceleration of the ball is only due to gravity, which is downward.

Thus, the velocity and acceleration of the ball at are 20√2 m/s and 10 m/s² respectively.

The distance traveled by a freely falling body in time can be calculated using the equation for uniformly accelerated motion: , where is the acceleration due to gravity. For this problem, we focus on the distance traveled during each specific second, rather than the total distance.

To find the distance fallen in each second:

• First Second: The distance traveled, .
• Second Second: Total distance at 2 seconds, . Calculate distance for the second second: .
• Third Second: Total distance at 3 seconds, . Calculate distance for the third second: .
• Fourth Second: Total distance at 4 seconds, . Calculate distance for the fourth second: .

Now, the distance traveled each second is .

The ratio of these distances is: .

Therefore, the ratio of the distances traveled by the body in the 1^st, 2^nd, 3^rd, and 4^th seconds is 1:3:5:7.

• f_s is the static frictional force.
• μ_s is the coefficient of static friction (given as 0.15).
• N is the normal force.

To determine the total length of the block, we analyze the bullet's motion. Initially, the bullet is fired with velocity and travels a distance of 24 cm through the block while its velocity decreases to . We use the equation of motion:

where , , , and is the acceleration (which is negative as the bullet is slowing down).

Now, we find the additional distance the bullet travels before coming to rest:

Hence, the bullet penetrates an additional 3 cm beyond the initial 24 cm to come to rest at the end of the block. Therefore, the total length of the block is:

The average speed of a vehicle is calculated as the total distance traveled divided by the total time taken. Since the vehicle travels half the distance with speed and the remaining half with speed , let's denote the total distance as . This means it travels at each speed.

1. Calculate Time Taken for Each Half:
- Time for first half:
- Time for second half:

2. Total Time Taken:

3. Simplify Total Time:

4. Average Speed Formula:

5. Simplify Average Speed:

Thus, the average speed of the vehicle is .