Question

A car with an initial speed of 22.1 km/h accelerates at a uniform rate of 0.84 m/s2 for 4.4 s. Find the final speed of the car. Answer in units of m/s. Find the displacement of the car after that time. Answer in units of km.

(Desperately need help with Part 2)

Answer 1

Final speed of car = 9.834 m/s

Displacement = 0.035 km

Step-by-step explanation:

Part 1

Final velocity, v of an object moving with an acceleration a m/s² for t seconds and an initial velocity u m/s is given by the formula
v = u + at m/s

`\textrm{Initial velocity } u = 22.1 km/h\\\\\textrm{Convert this to m/s:}\\\\\textrm{1 km/h = 1000m/3600 s}\\\\ 22.1 km/h = 22.1 x 1000/3600 = 6.139 m/s\\\\\boxed{\textrm{Final velocity } v = u + at = 6.138 + 0.84 x 4.4 = 9.834 m/s}`

Displacement s in m for a constant acceleration of a m/s² for a time t seconds is given by the formula:

`x = ut+(1)/(2)at^2`
where u is initial velocity

Plugging in the values we get

`s = 6.139\cdot 4.4 + (1)/(2)\cdot0.84\cdot (4.4)^2 = 35.143 \;\mathrm {meters}`
35.143 meters = 35.143/1000 = 0.035 km

Answer 2

1) 9.83 m/s (2 d.p.)

2) 0.04 km (2 d.p.)

Step-by-step explanation:

Constant Acceleration Equations (SUVAT)

`\boxed{\begin{array}{c}\begin{aligned}v&=u+at\\\\s&=ut+(1)/(2)at^2\\\\ s&=\left((u+v)/(2)\right)t\\\\v^2&=u^2+2as\\\\s&=vt-(1)/(2)at^2\end{aligned}\end{array}} \quad \boxed{\begin{minipage}{4.6 cm}$s$ = displacement in m\\\\$u$ = initial velocity in ms$^(-1)$\\\\$v$ = final velocity in ms$^(-1)$\\\\$a$ = acceleration in ms$^(-2)$\\\\$t$ = time in s (seconds)\end{minipage}}`

When using SUVAT, assume the object is modeled as a particle and that acceleration is constant.

Part 1

Given:

• u = 22.1 km/h
• a = 0.84 m/s²
• t = 4.4 s

Convert km/h into m/s by dividing by 3.6:

`\implies \sf 22.1\;km/h=(22.1)/(3.6)\;m/s`

Substitute the given values into the formula and solve for v:

`\begin{aligned}v&=u+at\\\implies v & = (22.1)/(3.6)+0.84(4.4)\\v & = 6.1388...+3.696\\v & = 9.834888...\\v & = 9.83 \sf \;m/s\;(2\:d.p.)\end{aligned}`

Therefore, the final speed of the car was 9.83 m/s (2 d.p.).

Part 2

Substitute the given values into the formula and solve for s, remembering to use the initial velocity in m/s:

`\begin{aligned}s & = ut+(1)/(2)at^2\\\implies s &=(22.1)/(3.6)(4.4)+(1)/(2)(0.84)(4.4)^2\\s &=27.0111...+8.1312\\s &=35.1423111...\sf m\end{aligned}`

Convert meters into kilometers by dividing by 1000:

`\implies (35.1423111...)/(1000)=0.0351423111...=0.04 \sf \; km\;(2\:d.p.)`

Therefore, the displacement of the car after 4.4 s was 0.04 km (2 d.p.).