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A particle is moving along the `x`-axis. Its velocity `v` at time `t` is given by
`v = (t^2+4)/sqrt(3t+1)` metres per second
Find the acceleration of the particle when `t = 2`.
Express your answer as an exact value in its simplest form. (3 marks)
--- 6 WORK AREA LINES (style=lined) ---
` (16sqrt(7))/49\ \ text(ms)^(−2)`
`v = (t^2+4)/sqrt(3t+1)`
| `alpha` | `= (dv)/(dt)` |
`text(Using quotient rule:)`
| `u=t^2+4,` | `v=(3t+1)^(1/2)` | |
| `u^{′} = 2t,` | `v^{′} = 3/2 (3t+1)^(-1/2)` | |
| `alpha` | `= (u^{′} v-v^{′} u)/v^2` |
| `= (2t (3t+1)^(1/2)-3/2(t^2+4) (3t+1)^(-1/2))/(3t+1)` |
`text(When)\ \ t = 2,`
| `alpha` | `= (4(7)^(1/2)-12(7)^(-1/2))/(7)` |
| `= (4sqrt(7))/7 -12/(7sqrt(7))` | |
| `= (28sqrt(7))/49-(12sqrt(7))/49` | |
| `= (16sqrt(7))/49\ \ text(ms)^(−2)` |
A particle is moving along the `x`-axis. Its velocity `v` at time `t` is given by
`v = sqrt(20t-2t^2)` metres per second
Find the acceleration of the particle when `t = 4`.
Express your answer as an exact value in its simplest form. (3 marks)
--- 6 WORK AREA LINES (style=lined) ---
`sqrt3/6\ \ text(ms)^(−2)`
`v = sqrt(20t-2t^2)`
| `a` | `= (dv)/(dt)` |
| `= 1/2 · (20t-2t^2)^(−1/2) · (20-4t)` |
`text(When)\ \ t = 4:`
| `a` | `= 1/2(20 · 4-2 · 4^2)^(−1/2)(20-16)` |
| `= 2/(sqrt48)` | |
| `= 2/(4sqrt3) xx sqrt3/sqrt3` | |
| `= sqrt3/6\ \ text(ms)^(−2)` |