Question

A sports car accelerates approximately as shown in the velocity-time graph of Fig. 2-43. (The short flat spots in the curve represent manual shifting of the gears.) Estimate the car’s average acceleration in (a) second gear and (b) fourth gear.

FIGURE 2-43Problem 56. The velocity of a car as a function of time, starting from a dead stop. The flat spots in the curve represent gear shifts.

Short answer

(a) The car’s average acceleration in the second gear is $2.5{\mathrm{ms}}^{-2}$.

(b) The car’s average acceleration in the fourth gear is $0.73{\mathrm{ms}}^{-2}$.

Step-by-step solution

Step 1. Average acceleration

When the velocity of an object during its motion changes by unequal amounts in equal time periods, the acceleration of the object is variable. Therefore, there is a need to calculate the average acceleration of the object.

Average acceleration of the object in a given time period is calculated by taking the ratio of the change in velocity to the total time taken, i.e.,

$\begin{array}{c}\mathrm{Average}\mathrm{acceleration}=\frac{\mathrm{Total}\mathrm{change}\mathrm{in}\mathrm{velocity}}{\mathrm{Total}\mathrm{time}\mathrm{taken}}\\ \overline{a}=\frac{\Delta v}{\Delta t}\end{array}$

You can also calculate the average acceleration of the object using the velocity-time graph. The slope of the straight line formed by joining the two points on the velocity-time graph gives the average acceleration of the object between these points.

Step 2. (a) Determination of the average acceleration of the car in the second gear

It can be estimated from the graph that the velocity of the sports car in second gear changes from $13{\mathrm{ms}}^{-1}$to $23{\mathrm{ms}}^{-1}$when the time changes from 4 s to 8 s.

The slope of the velocity-time graph during this interval gives the average acceleration of the car.

Thus, the average acceleration of the car in the second gear can be calculated as shown below.

$\begin{array}{c}\overline{a}=\frac{\Delta v}{\Delta t}\\ =\frac{\left(23-13\right){\mathrm{ms}}^{-1}}{\left(8-4\right)\mathrm{s}}\\ =\frac{10}{4}{\mathrm{ms}}^{-2}\\ =2.5{\mathrm{ms}}^{-2}\end{array}$

So, the car’s average acceleration in the second gear is $2.5{\mathrm{ms}}^{-2}$.

Step 3. (b) Determination of the average acceleration of the car in the fourth gear

It can be estimated from the graph that the velocity of the sports car in the fourth gear changes from $36{\mathrm{ms}}^{-1}$to $44{\mathrm{ms}}^{-1}$when the time changes from 16 s to 27 s. The slope of the velocity-time graph during this interval gives the average acceleration of the car.

Thus, the average acceleration of the car in the fourth gear can be calculated as

shown below.

$\begin{array}{c}\overline{a}=\frac{\Delta v}{\Delta t}\\ =\frac{\left(44-36\right){\mathrm{ms}}^{-1}}{\left(27-16\right)\mathrm{s}}\\ =\frac{8}{11}{\mathrm{ms}}^{-2}\\ =0.73{\mathrm{ms}}^{-2}\end{array}$

So, the car’s average acceleration in the fourth gear is $0.73{\mathrm{ms}}^{-2}$.