linear motion Questions and Answers - TopperLearning - Free Printable

Here are the step-by-step solutions for the three problems shown in the image.

Problem 4: Identify the velocity-time graph

Step 1: Analyze the first part of the motion.
The problem states the car starts from rest with a constant acceleration $a_1$.
* "Starts from rest" means initial velocity is 0. The graph must start at the origin $(0,0)$.
* "Constant acceleration" means the velocity increases at a steady rate. On a velocity-time graph, constant acceleration is represented by a straight line with a positive slope (going up).
* This eliminates graphs (2) and (3) because they show curved lines, which would mean changing acceleration.

Step 2: Analyze the second part of the motion.
The car then retards (slows down) at a constant rate $a_2$ until it stops.
* "Constant retardation" means the velocity decreases at a steady rate. This is represented by a straight line with a negative slope (going down).
* The car eventually stops, so the final velocity is 0. The graph should end on the time axis ($t$-axis).

Conclusion:
Graph (1) shows a straight line going up to a peak velocity $v_0$, followed by a straight line going down to zero. This matches the description perfectly.

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Problem 5: Relation between accelerations and times

Step 1: Define the variables.
* $t_1$: Time taken for acceleration.
* $t_2$: Time taken for retardation.
* $a_1$: Acceleration.
* $a_2$: Retardation (magnitude of deceleration).
* $v_0$: Maximum velocity reached.

Step 2: Write equations for both phases.
* Phase 1 (Acceleration): The car starts from 0 and reaches $v_0$ in time $t_1$.
Using the formula $v = u + at$:
$$v_0 = 0 + a_1 t_1 \implies v_0 = a_1 t_1$$

* Phase 2 (Retardation): The car starts at $v_0$ and slows down to 0 in time $t_2$.
Here, the final velocity is 0, initial is $v_0$, and acceleration is $-a_2$ (since it's slowing down).
$$0 = v_0 - a_2 t_2 \implies v_0 = a_2 t_2$$

Step 3: Equate the expressions for $v_0$.
Since both expressions equal the maximum velocity $v_0$, we can set them equal to each other:
$$a_1 t_1 = a_2 t_2$$

This matches option (2).

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Problem 6: Relation between $v_0$, $a_1$, $a_2$, and total time $t_0$

Step 1: Express individual times in terms of $v_0$.
From Problem 5, we know:
$$t_1 = \frac{v_0}{a_1}$$
$$t_2 = \frac{v_0}{a_2}$$

Step 2: Use the total time.
The total time of the journey is $t_0$. Therefore:
$$t_0 = t_1 + t_2$$

Substitute the expressions from Step 1 into this equation:
$$t_0 = \frac{v_0}{a_1} + \frac{v_0}{a_2}$$

Step 3: Solve for $v_0$.
Factor out $v_0$ on the right side:
$$t_0 = v_0 \left( \frac{1}{a_1} + \frac{1}{a_2} \right)$$

Find a common denominator for the terms inside the parenthesis:
$$t_0 = v_0 \left( \frac{a_2 + a_1}{a_1 a_2} \right)$$

Now, isolate $v_0$ by multiplying both sides by the reciprocal of the fraction:
$$v_0 = t_0 \left( \frac{a_1 a_2}{a_1 + a_2} \right)$$

Rearranging it to match the options:
$$v_0 = \left( \frac{a_1 a_2}{a_1 + a_2} \right) t_0$$

This matches option (1).

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Final Answer:
4. (1)
5. (2)
6. (1)