A car is running at 15m/s when the driver decides to accelerate in order to reach a speed of 45m/s over a 200m. (a) calculate the time (b) the acceleration
To calculate the time:
Using the equation of motion:
\(v = u + at\),
where:
v = final velocity = 45 m/s,
u = initial velocity = 15 m/s,
a = acceleration,
t = time.
Rearranging the equation:
\(t = \frac{{v - u}}{a}\).
Given the values:
\(v = 45\) m/s,
\(u = 15\) m/s.
To find the time, we need to calculate the acceleration (a) first.
To calculate the acceleration:
We can use the equation of motion:
\(v^2 = u^2 + 2as\),
where:
v = final velocity = 45 m/s,
u = initial velocity = 15 m/s,
a = acceleration,
s = distance traveled = 200 m.
Rearranging the equation:
\(a = \frac{{v^2 - u^2}}{2s}\).
Given the values:
\(v = 45\) m/s,
\(u = 15\) m/s,
\(s = 200\) m.
(a) Calculating the acceleration:
\(a = \frac{{45^2 - 15^2}}{2 \cdot 200}\).
\(a = \frac{{2025 - 225}}{400}\).
\(a = \frac{{1800}}{400}\).
\(a = 4.5 \, \text{m/s}^2\).
(b) Calculating the time:
\(t = \frac{{45 - 15}}{4.5}\).
\(t = \frac{{30}}{4.5}\).
\(t = 6.67\) seconds (approximately).
Therefore, the time taken to reach a speed of 45 m/s is approximately 6.67 seconds, and the acceleration is 4.5 m/s^2.
To calculate the time it takes for the car to reach a speed of 45 m/s, we can use the formula:
t = (v - u) / a
where:
t = time (unknown)
v = final velocity = 45 m/s
u = initial velocity = 15 m/s
a = acceleration (unknown)
Substituting the given values into the formula:
t = (45 - 15) / a
t = 30 / a
Now, to find the acceleration, we need to use the second equation of motion:
v^2 = u^2 + 2as
where:
v = final velocity = 45 m/s
u = initial velocity = 15 m/s
a = acceleration (unknown)
s = distance = 200 m
Substituting the given values:
45^2 = 15^2 + 2a * 200
2025 = 225 + 400a
Rearranging the equation:
1800 = 400a
a = 1800 / 400
a = 4.5 m/s^2
Now we can substitute the value of a back into the time equation:
t = 30 / a
t = 30 / 4.5
t = 6.67 seconds (rounded to two decimal places)
Therefore, the car will take approximately 6.67 seconds to reach a speed of 45 m/s, and the acceleration required is 4.5 m/s^2.
To calculate the time and acceleration, we can use the equations of motion.
The first equation is:
v = u + at
where:
v = final velocity
u = initial velocity
a = acceleration
t = time
We know the initial velocity (u) is 15 m/s and the final velocity (v) is 45 m/s.
(a) To calculate the time (t), we rearrange the equation to solve for t:
t = (v - u) / a
In this case, we need the value of acceleration (a) in order to calculate time.
The second equation is:
s = ut + (1/2)at²
where:
s = distance
u = initial velocity
t = time
a = acceleration
We know the initial velocity (u) is 15 m/s, the final velocity (v) is 45 m/s, and the distance (s) is 200 m.
We can substitute these values into the equation, which becomes:
200 = 15t + (1/2)a * t²
We have two unknowns, time (t) and acceleration (a). To solve for both variables, we need another equation.
The third equation is:
v² = u² + 2as
We know the initial velocity (u) is 15 m/s, the final velocity (v) is 45 m/s, and the distance (s) is 200 m.
We can substitute these values into the equation, which becomes:
45² = 15² + 2a * 200
Now we have two equations with two unknowns:
Equation 1: 200 = 15t + (1/2)a * t²
Equation 2: 45² = 15² + 2a * 200
We can solve these equations simultaneously to find the values of time (t) and acceleration (a).