Kinematics of motion in a straight line — practice questions

First, list the known SUVAT variables. We are given: $u = 12 \text{ m s}^{-1}$ $a = 1.5 \text{ m s}^{-2}$ $t = 8 \text{ s}$

We need to find the final velocity, $v$, and the displacement, $s$.

To find v: Choose the equation that links u, a, t, and v: $v = u + at$. $v = 12 + (1.5)(8)$ $v = 12 + 12$ $v = 24 \text{ m s}^{-1}$

To find s: We can use an equation linking u, a, t, and s: $s = ut + \frac{1}{2}at^2$. $s = (12)(8) + \frac{1}{2}(1.5)(8^2)$ $s = 96 + \frac{1}{2}(1.5)(64)$ $s = 96 + 48$ $s = 144 \text{ m}$

The final velocity is 24 m s⁻¹ and the distance travelled is 144 m.

The graph is a straight line, which means the acceleration is constant.

To find acceleration, a: Acceleration is the gradient of the v-t graph. $a = \frac{\text{change in velocity}}{\text{change in time}} = \frac{v_2 - v_1}{t_2 - t_1}$ $a = \frac{12 - 4}{10 - 0} = \frac{8}{10}$ $a = 0.8 \text{ m s}^{-2}$

To find distance travelled, s: Distance travelled is the area under the v-t graph. The shape is a trapezium. Area = $\frac{1}{2}(a+b)h$, where a and b are the parallel sides (the initial and final velocities) and h is the height (the time interval). $s = \frac{1}{2}(4 + 12) \times 10$ $s = \frac{1}{2}(16) \times 10$ $s = 8 \times 10 = 80 \text{ m}$

The cyclist's acceleration is 0.8 m s⁻² and the total distance travelled is 80 m.

We are given $a = 6t - 2$.

To find velocity, v: Integrate acceleration with respect to time. $v = \int a \, dt = \int (6t - 2) \, dt$ $v = 3t^2 - 2t + C$ We are given that when $t=0$, $v=3$. Substitute these values to find C: $3 = 3(0)^2 - 2(0) + C \implies C = 3$ So, the expression for velocity is $v = 3t^2 - 2t + 3$ m s⁻¹.

To find displacement, s: Integrate velocity with respect to time. $s = \int v \, dt = \int (3t^2 - 2t + 3) \, dt$ $s = t^3 - t^2 + 3t + D$ We are given that when $t=0$, the particle is at the origin, so $s=0$. Substitute these values to find D: $0 = (0)^3 - (0)^2 + 3(0) + D \implies D = 0$ So, the expression for displacement is $s = t^3 - t^2 + 3t$ m.