A particle accelerating uniformly at 50 centimeters per second squared was moving in a straight line. If its initial velocity was 45 kilometers per hour in the same direction as the acceleration, find the time required for it to cover 54 meters.
In order to answer this question, we will use the equations of constant acceleration, otherwise known as the SUVAT equations. 𝑠 is the displacement, 𝑢 the initial velocity, 𝑣 the final velocity, 𝑎 the acceleration, and 𝑡 the time. In this question, we are told that the uniform acceleration is 50 centimeters per second squared. The initial velocity is 45 kilometers per hour. We want to calculate the time 𝑡 that it takes to cover a distance of 54 meters. The displacement is 54 meters.
The equation we will use in this question is 𝑠 equals 𝑢𝑡 plus a half 𝑎𝑡 squared. We do have a problem in this question, however, as our units are different. We have distance or displacement measured in meters, centimeters, and kilometers and time measured in seconds and hours. There are 100 centimeters in one meter and 1,000 meters in one kilometer. This means that 54 meters is equal to 5,400 centimeters. Multiplying 45 by 1,000 and then by 100 gives us 4,500,000. This means that 45 kilometers per hour is the same as 4,500,000 centimeters per hour. We recall that there are 60 seconds in one minute and 60 minutes in one hour. This means that we need to divide 4,500,000 by 60 and then by 60 again.
We can therefore say that the initial velocity is equal to 1,250 centimeters per second. As our units of length or distance are now centimeters and our units of time are seconds, we can substitute our values into the equation. This gives us 5,400 is equal to 1,250𝑡 plus a half multiplied by 50𝑡 squared. A half of 50 is 25, so the last term becomes 25𝑡 squared. We can then subtract 5,400 from both sides of the equation. All three terms have a common factor of 25. So dividing by 25 gives us 𝑡 squared plus 50𝑡 minus 216 is equal to zero.
Our quadratic expression will factor or factorize into two pairs of parentheses or brackets. The first term is 𝑡, as 𝑡 multiplied by 𝑡 is 𝑡 squared. 54 multiplied by four is 216. This means that 54 multiplied by negative four is negative 216. And as 54 minus four is 50, our two parentheses are 𝑡 plus 54 and 𝑡 minus four. This gives us two possible solutions. Either 𝑡 plus 54 equals zero or 𝑡 minus four equals zero. Solving these equations gives us values of 𝑡 of negative 54 and four. As time cannot be negative, the correct answer is 𝑡 equals four.
We can therefore conclude that the time taken for the particle to cover 54 meters is four seconds.