4) A moving body decelerate if \( \qquad \) (a) \( \hat{s} \), à have same direction. (b) \( \bar{s}, \vec{a} \) act in opposite directions. (c) \( \bar{v}, \bar{a} \) have same direction. (d) \( \stackrel{\rightharpoonup}{v}, \vec{a} \) act in opposite directions. 5) The speed of a moving body is given by the relation \( v=31+t^{2} \), then the acceleration of the body after 4 seconds from the beginning of the motion equals \( \qquad \) (a) 4 (b) 11 (c) 26 (d) 28 6) A body moves in a striaght line then the relation between its position vector and time could be \( \qquad \) ... (a) \( \vec{x}=(3 t+4) \vec{c} \) (b) \( \bar{x}=(\sin t) \bar{c} \) (c) \( \vec{x}=[2+\ln (t+1)] \vec{c} \) (d) all the previous 7)Which of the following position vectors is considered a displacement vector of a moving particle? (a) \( \vec{x}=\left(3 t^{2}-5\right) \vec{c} \) (b) \( \vec{x}=(3 t+1) \vec{c} \) (c) \( \vec{x}=\left(\frac{3 t-2}{t+1}\right) \vec{c} \) (d) \( \vec{x}=\left((1-2)^{2}-4\right) \vec{c} \)

Let's solve each of the problems step by step. ### Problem 4 A moving body decelerates if: - (a) \( \hat{s} \), à have same direction. - (b) \( \bar{s}, \vec{a} \) act in opposite directions. - (c) \( \bar{v}, \bar{a} \) have same direction. - (d) \( \stackrel{\rightharpoonup}{v}, \vec{a} \) act in opposite directions. **Explanation:** - Deceleration occurs when the velocity vector \( \vec{v} \) and the acceleration vector \( \vec{a} \) are in opposite directions. This means that the body is slowing down. - Therefore, the correct answer is **(d)** \( \stackrel{\rightharpoonup}{v}, \vec{a} \) act in opposite directions. ### Problem 5 The speed of a moving body is given by the relation \( v = 31 + t^2 \). We need to find the acceleration after 4 seconds. **Step 1: Find the acceleration.** Acceleration \( a \) is the derivative of velocity \( v \) with respect to time \( t \): \[ a = \frac{dv}{dt} \] **Step 2: Differentiate \( v \).** \[ v = 31 + t^2 \implies \frac{dv}{dt} = 2t \] **Step 3: Calculate \( a \) at \( t = 4 \).** \[ a = 2(4) = 8 \] However, it seems I made a mistake in the options provided. Let's check the options again: - (a) 4 - (b) 11 - (c) 26 - (d) 28 Since \( a = 8 \) is not an option, let's check the calculations again. The acceleration is indeed \( 2t \), and at \( t = 4 \), it should be \( 8 \). It seems there might be a misunderstanding in the options provided. The correct acceleration after 4 seconds is \( 8 \). ### Problem 6 A body moves in a straight line, and we need to determine which relation between its position vector and time is valid. - (a) \( \vec{x} = (3t + 4) \vec{c} \) is linear in \( t \). - (b) \( \bar{x} = (\sin t) \bar{c} \) is periodic and valid. - (c) \( \vec{x} = [2 + \ln(t + 1)] \vec{c} \) is valid as well. **Conclusion:** All the given options represent valid relations for a position vector in terms of time. Therefore, the answer is **(d)** all the previous. ### Problem 7 We need to determine which of the following position vectors is considered a displacement vector of a moving particle. - (a) \( \vec{x} = (3t^2 - 5) \vec{c} \) is a quadratic function of \( t \). - (b) \( \vec{x} = (3t + 1) \vec{c} \) is linear in \( t \). - (c) \( \vec{x} = \left(\frac{3t - 2}{t + 1}\right) \vec{c} \) is a rational function of \( t \). - (d) \( \vec{x} = \left((1 - 2)^2 - 4\right) \vec{c} \) is a constant vector. **Conclusion:** Options (a), (b), and (c) represent valid displacement vectors as they depend on time \( t \). Option (d) is a constant and does not represent a displacement vector. Therefore, the correct answers are **(a)**, **(b)**, and **(c)**. If you have any further questions or need clarification on any of the problems, feel free to ask!