For motion in a straight line with constant acceleration, we derived two primary equations given by Eq. (4.4a) and (4.4b). Using these two equations, three more equations can be derived, out of which we derived one given in Eq. (4.4c). Derive the remaining two equations given below: s = vt − ½at² and s = ½(u + v)t. In mathematics, you have learnt the formula for calculating the area of a trapezium. Using that formula, derive the second equation given above.
Derive the equation s = vt − ½at² from the primary kinematic equations.
We start with the primary kinematic equations for motion with constant acceleration: Eq. (4.4a) v = u + at and Eq. (4.4b) s = ut + ½at². To derive the required equation, we eliminate the initial velocity 'u' from Eq. (4.4b). From Eq. (4.4a), we express 'u' as u = v - at. Substituting this into Eq. (4.4b), we get s = (v - at)t + ½at². Expanding this, we obtain s = vt - at² + ½at². Combining the terms with 't²', we get s = vt - ½at².
Derive the equation s = ½(u + v)t using the formula for the area of a trapezium.
To derive the second equation, we use the velocity-time graph for an object moving with constant acceleration. The displacement of the object is equal to the area under the velocity-time graph. For constant acceleration, this graph is a straight line, and the area under it forms a trapezium. The two parallel sides of this trapezium represent the initial velocity 'u' and the final velocity 'v'. The height of the trapezium represents the time interval 't'. Using the mathematical formula for the area of a trapezium, Area = ½ × (sum of parallel sides) × height. Substituting the physical quantities, we get Area = ½ × (u + v) × t. Since the area under the v-t graph represents displacement, we get s = ½(u + v)t.
Explanation
Part (a) requires algebraic substitution of 'u' from the first kinematic equation into the second to eliminate the initial velocity. Part (b) uses the graphical interpretation of displacement as the area under the velocity-time graph, which forms a trapezium for uniformly accelerated motion, allowing the application of the geometric area formula.