how long will it take a car to cover a distance of 500 meters if it accelerates from rest at a rate of 5.50 m/s2?
s = 1/2 at^2
500 = 1/2 * 5.5 t^2
61
To determine how long it will take for a car to cover a distance of 500 meters while accelerating at a rate of 5.50 m/s^2, we can use the equations of motion. Specifically, we will use the equation:
\[s = ut + \frac{1}{2}at^2\]
where:
- \(s\) is the distance covered (500 meters),
- \(u\) is the initial velocity (0 m/s as the car is at rest),
- \(a\) is the acceleration (5.50 m/s^2), and
- \(t\) is the time taken (what we want to find).
Rearranging the equation, we get:
\[t^2 + \frac{2u}{a}t - \frac{2s}{a} = 0\]
Substituting the known values:
\[t^2 + \frac{2(0)}{5.50}t - \frac{2(500)}{5.50} = 0\]
\[t^2 - \frac{2000}{5.50} = 0\]
Simplifying further,
\[t^2 - 363.64 = 0\]
To solve this quadratic equation, we can either use the quadratic formula or factor the equation. In this case, factoring is more straightforward. The equation can be factored as:
\[(t + 19.09)(t - 19.09) = 0\]
Setting each factor to zero, we get:
\(t + 19.09 = 0\)
\(t - 19.09 = 0\)
Solving for \(t\) in each equation, we find:
\(t = -19.09\) (discarded as time cannot be negative)
\(t = 19.09\) seconds
Therefore, it will take the car approximately 19.09 seconds to cover a distance of 500 meters while accelerating at a rate of 5.50 m/s^2.