Average Speed -2

How to analyse the problem: The speeds are unequal: v₁ and v₂, while the time intervals are equal: t, t.

How to crack it: Start from average speed = total distance ÷ total time. Since the time intervals are equal, find each distance using d = vt.

Therefore, for equal time intervals:

Conclusion: When the time intervals are equal, the average speed is the arithmetic mean of the two speeds.

Given:

First speed = v₁
Second speed = v₂
Both time intervals = t

Step 1: Find the distances

Step 2: Find total distance

Step 3: Find total time

Step 4: Calculate average speed

Cancel the common factor t:

How to analyse the problem: The car moves with the same speed v during both time intervals. Since the speed does not change, the average speed must also be v.

How to crack it: Start with the basic definition of average speed:

Basic formula:

Since the speed is v:

Derived formula:

Conclusion: When the speed remains the same, the average speed is equal to that constant speed, regardless of the equal time intervals.

Given:

Distance covered in the first interval:

Distance covered in the second interval:

Therefore,

Taking v and t common:

How to analyse the problem: The car starts from rest, so its initial speed is zero. Since the car accelerates uniformly, its velocity increases at a constant rate.

How to crack it: For uniformly accelerated motion, the average speed is the arithmetic mean of the initial and final speeds. First find the final speed using the equation of motion.

Basic formulas:

Since the car starts from rest:

Therefore, the formulas become:

Conclusion: For a body starting from rest and moving with uniform acceleration, the average speed during time t is at/2.

Given:

First, find the final speed:

Now calculate the average speed:

How to analyse the problem: The car starts from rest, so its initial speed is zero. However, the acceleration is non-uniform, meaning the acceleration is not constant throughout the motion.

How to crack it: When acceleration is non-uniform, we cannot directly use the uniform-acceleration formula v_avg = (u + v)/2. Instead, always return to the definition of average speed.

Basic formula:

Since the car starts from rest:

But the condition of starting from rest alone is not enough to determine the average speed. The distance travelled must be known or determined from the actual motion.

Derived form:

Conclusion: For non-uniform acceleration, use the fundamental definition of average speed: total distance divided by total time. The shortcut (u + v)/2 cannot be used unless the acceleration is uniform.

The car starts from rest, so:

But the acceleration is non-uniform. Therefore, the uniform-acceleration relation v_avg = (u + v)/2 is not generally valid.

We must use the definition of average speed:

Therefore, if the total distance travelled is s during time t:

How to analyse the problem: The car starts from rest, but its acceleration is non-uniform. Therefore, we should not use the uniform-acceleration shortcut v_avg = (u + v)/2.

How to crack it: When the motion is non-uniformly accelerated, divide the motion into the given intervals, find the total distance and total time, and then use the definition of average speed.

Basic formula:

For two intervals:

Conclusion: For non-uniform acceleration, average speed is determined from total distance and total time. The initial condition of rest does not make the uniform-acceleration formula applicable.

Given:

Total distance travelled:

Total time:

Therefore, the average speed is:

How to analyse the problem: The car starts from rest, but its acceleration is not constant. Instead, acceleration is given as a function of time.

How to crack it: For non-uniform acceleration, do not use v_avg = (u + v)/2. First obtain velocity from acceleration and then obtain displacement from velocity.

Basic formulas:

Therefore, when a(t) is known:

Finally, average speed is obtained from:

Conclusion: When acceleration is a function of time, use integration to find velocity and displacement. Then use total distance divided by total time to obtain average speed.

Given:

Since a = dv/dt:

Integrating from 0 to t:

Now find the distance travelled:

Therefore:

How to analyse the problem: The car already has an initial speed u, and then it accelerates uniformly. Therefore, both the initial and final speeds are different.

How to crack it: First find the final speed using the first equation of motion. Since acceleration is uniform, the average speed is the arithmetic mean of the initial and final speeds.

Basic formulas:

Substituting v = u + at into the average-speed formula:

Conclusion: For initial velocity u and uniform acceleration a acting for time t, the average speed is u + at/2.

Given:

First, find the final speed:

Since the acceleration is uniform:

How to analyse the problem: The car is initially moving with speed u and its speed decreases uniformly due to retardation. Since the retardation is uniform, the acceleration is constant and negative.

How to crack it: Treat retardation as negative acceleration. First find the final speed using the equation of motion. Then use the average-speed formula for uniformly accelerated motion.

Basic formulas:

For retardation of magnitude r:

For uniform retardation, the average speed is:

Substituting v = u - rt:

Conclusion: During uniform retardation, the average speed is the arithmetic mean of the initial and final speeds, provided the object continues moving in the same direction throughout the interval.

Given:

Retardation is opposite to the direction of motion, so:

Find the final speed:

Now calculate the average speed:

How to analyse the problem: The car starts from rest and moves with uniform acceleration. The distance and time are directly given, so average speed can be found immediately from its definition.

How to crack it: Do not unnecessarily find acceleration or final velocity. When total distance and total time are already given, directly use total distance divided by total time.

Basic formula:

For distance s covered in time t:

Since the car starts from rest and acceleration is uniform:

Therefore:

Conclusion: When distance and time are given, average speed is simply distance divided by time. The uniform-acceleration condition provides additional information but is not needed for this direct calculation.

Given:

Average speed is:

How to analyse the problem: The initial speed and final speed are given directly. Since the acceleration is uniform, the velocity changes uniformly with time.

How to crack it: When initial and final speeds are known and acceleration is uniform, there is no need to find acceleration, distance, or time. Directly take the arithmetic mean of the initial and final speeds.

Basic formula:

This result follows from uniform acceleration because the velocity changes linearly with time.

Using the equation:

The average speed can also be written as:

Conclusion: For uniform acceleration, when the initial and final speeds are given, average speed is the arithmetic mean of those two speeds.

Given:

Since the acceleration is uniform, use: