Question

Consider a particle moving along the \(x\) -axis where \(x(t)\) is the position of the particle at time \(t, x^{\prime}(t)\) is its velocity, and \(x^{\prime \prime}(t)\) is its acceleration. Acceleration The maker of an automobile advertises that it takes 13 seconds to accelerate from 25 kilometers per hour to 80 kilometers per hour. Assuming constant acceleration, compute the following. (a) The acceleration in meters per second per second (b) The distance the car travels during the 13 seconds

Short answer

The acceleration of the car is approximately 1.176 m/s^2 and the car travels approximately 157.87 meters during the 13 seconds.

Step-by-step solution

Conversion of Units

Convert the speeds from kilometers/hour to meters/second. Multiply by 1000 to go from kilometer to meter and divide by 3600 to change hour into second. Thus, the initial speed \(v_1 = 25 * 1000 / 3600 = 6.944 m/s\) and final speed \(v_2 = 80 * 1000 / 3600 = 22.222 m/s\). The time \(t = 13 s\).

Calculate Acceleration

Acceleration is defined as the change in velocity with respect to time. Using the formula \(a = (v_2 - v_1) / t\), the acceleration \(a = (22.222 - 6.944) / 13 = 1.176 m/s^2\).

Calculate Distance

For this, the equation of motion can be used: \(s = v_1*t + 0.5*a*t^2\), where \(s\) is the distance travelled. By plugging in the known values into the equation, the distance \(s = 6.944 * 13 + 0.5 * 1.176 * (13^2) = 157.87 m\).