All questions of Translational Motion and Calculations for MCAT Exam

Understanding the Motion of a Falling Object with Air Resistance
When an object is dropped from a great height, such as the Empire State Building, its motion is influenced significantly by air resistance. Here's a detailed breakdown of what happens during its fall.
Initial Free Fall
- When the object is initially dropped, it accelerates downward due to gravity, which is approximately 9.81 m/s².
- During this phase, the object’s speed increases rapidly as it falls.
Effect of Air Resistance
- As the object gains speed, it encounters air resistance, which acts in the opposite direction to its motion.
- This resistance increases with the speed of the object; hence, the faster the object falls, the greater the air resistance acting against it.
Transition to Terminal Velocity
- Eventually, the force of air resistance grows enough to balance the gravitational force acting on the object.
- At this point, the object stops accelerating and continues to fall at a constant speed known as terminal velocity.
Acceleration Changes
- Initially, the object has a constant acceleration due to gravity.
- As air resistance increases, the net force acting on the object decreases, leading to a reduction in acceleration.
- The object’s acceleration decreases until it reaches zero at terminal velocity, where it moves at a constant speed.
Conclusion
- Therefore, option 'D' is correct: "Its acceleration will decrease until the object starts moving with a constant speed."
- This describes the transition from free fall to terminal velocity, illustrating how air resistance impacts the motion of the falling object.

Minimum Average Acceleration to Reach Orbital Altitude of Earth

To determine the minimum average acceleration required for a space shuttle to reach the orbital altitude of Earth, we need to use the given information of the shuttle's final velocity and the time it takes to reach that velocity.

Given:
Final velocity (v) = 8,000 m/s
Time (t) = 8.5 minutes = 8.5 * 60 seconds = 510 seconds

To find the minimum average acceleration, we can use the following equation of motion:

v = u + at

Where:
v = final velocity
u = initial velocity (which is 0 in this case)
a = acceleration
t = time

Step 1: Calculate the initial velocity (u)

Since the initial velocity (u) is given as 0, we can substitute this value into the equation:

v = u + at
8000 = 0 + a * 510

Step 2: Rearrange the equation and solve for acceleration (a)

8000 = a * 510
a = 8000 / 510
a ≈ 15.69 m/s^2

Since the question asks for the minimum average acceleration, we round the value to the nearest whole number, which is 16 m/s^2.

Answer: The minimum average acceleration required for the space shuttle to reach the orbital altitude of Earth is approximately 16 m/s^2. Therefore, the correct answer is option 'B' - 15 m/s^2.

Given data:
- Landing speed = 72 m/s
- Stopping distance = 2000 m

Calculating deceleration:
- Initial velocity (u) = 72 m/s
- Final velocity (v) = 0 m/s
- Distance (s) = 2000 m
- Using the equation of motion: v^2 = u^2 + 2as
- 0 = (72)^2 + 2*a*2000
- Solving for acceleration (a): a = -72^2 / (2*2000) = -1.296 m/s^2

Calculating time taken to stop:
- Using the equation of motion: v = u + at
- 0 = 72 + (-1.296)*t
- Solving for time (t): t = 72 / 1.296 ≈ 55.56 seconds
Therefore, it would take approximately 56 seconds for a Boeing 747 to come to a full stop after landing. So, the correct answer is option 'C' - 56 seconds.

Statement: The average speed is 37.5 miles per hour.

To determine the accuracy of this statement, let's examine the given information and calculate the average speed of the car.

Given information:
- Distance from point A to B: 150 miles
- Time taken to travel from A to B: 3 hours
- Time taken to return from B to A: 5 hours

Calculating average speed:
Average speed is calculated by dividing the total distance traveled by the total time taken.

Total distance traveled = Distance from A to B + Distance from B to A = 150 miles + 150 miles = 300 miles

Total time taken = Time taken from A to B + Time taken from B to A = 3 hours + 5 hours = 8 hours

Average speed = Total distance traveled / Total time taken = 300 miles / 8 hours = 37.5 miles per hour

Therefore, the statement that the average speed is 37.5 miles per hour is accurate.

Explanation:
The average speed of an object is defined as the total distance traveled divided by the total time taken. In this case, the car traveled a total distance of 300 miles (150 miles from A to B and 150 miles from B to A) in a total time of 8 hours (3 hours from A to B and 5 hours from B to A). Dividing the total distance by the total time gives us an average speed of 37.5 miles per hour.

It is important to note that average speed does not take into account the variations in speed during different parts of the journey. The car could have traveled at different speeds during different segments of the trip, but the average speed provides a single value that represents the overall speed of the car.

Therefore, option B is the most accurate statement as it correctly describes the average speed of the car as 37.5 miles per hour.

Explanation:
Acceleration can be calculated using the equation:
\[a = \frac{v_f - v_i}{t}\]
Where:
- \(a\) = acceleration
- \(v_f\) = final velocity
- \(v_i\) = initial velocity
- \(t\) = time
Given that the dragster accelerates from rest, the initial velocity is 0 m/s. The final velocity can be calculated using the equation for distance traveled with constant acceleration:
\[d = \frac{1}{2} a t^2\]

Calculations:
- Distance, \(d\) = 400 meters
- Time, \(t\) = 4 seconds
Using the distance equation:
\[400 = \frac{1}{2} a (4)^2\]
\[400 = 8a\]
\[a = \frac{400}{8} = 50 \, m/s^2\]
Therefore, the acceleration of the dragster would be 50 m/s^2 if it sped down a track in exactly 4 seconds.
This acceleration is the correct answer as option 'C'.

To home plate in Major League Baseball is 60 feet 6 inches (18.4 meters).

Understanding the Problem:
First, we need to understand the situation presented in the problem. A police car is chasing a minivan that is 600 meters ahead and traveling at a speed of 45 m/s. The police car accelerates quickly to 60 m/s in order to catch up with the minivan.

Calculating the Relative Speed:
In order to catch up with the minivan, the police car needs to cover the initial 600 meters between them. The relative speed between the police car and the minivan is 60 m/s - 45 m/s = 15 m/s.

Calculating the Time to Catch Up:
To calculate the time it will take for the police car to catch up with the minivan, we can use the formula:
\[ \text{Time} = \frac{\text{Distance}}{\text{Speed}} \]
\[ \text{Time} = \frac{600 \text{ meters}}{15 \text{ m/s}} \]
\[ \text{Time} = 40 \text{ seconds} \]
Therefore, it will take the police car 40 seconds to catch up with the minivan that is 600 meters ahead. So, the correct answer is option c) 40 seconds.

Calculating Average Acceleration
To calculate the average acceleration of the Thrust SSC during the 1-minute acceleration period, we can use the formula:
\[a = \dfrac{v_f - v_i}{t}\]
- \(a\) is the average acceleration
- \(v_f\) is the final velocity (760.34 mph)
- \(v_i\) is the initial velocity (0 mph)
- \(t\) is the time taken to reach the final velocity (1 minute or 60 seconds)

Converting Units
Before plugging the values into the formula, we need to convert the final velocity from mph to m/s:
\[760.34 \text{ mph} \times \dfrac{1609.344 \text{ m}}{1 \text{ mile}} \times \dfrac{1 \text{ hour}}{3600 \text{ seconds}} = 340.29 \text{ m/s}\]

Calculating Average Acceleration
Now, we can substitute the values into the formula:
\[a = \dfrac{340.29 \text{ m/s} - 0 \text{ m/s}}{60 \text{ s}} = \dfrac{340.29 \text{ m/s}}{60 \text{ s}} = 5.67 \text{ m/s}^2\]
Therefore, the average acceleration of the Thrust SSC during the 1-minute acceleration period was 5.67 m/s².