A car of mass 760kg travel with a velocity of 14.0ms-1 . If the brakes are applied , the car travels a distance of 55.0m before coming to rest . Calculate the retarding force on the car .
We can use the formula:
retarding force = (mass × change in velocity) ÷ time
We need to first find the time it took for the car to come to rest, which can be found using the equation:
final velocity = initial velocity + (acceleration × time)
Since the final velocity is 0 m/s (it's come to rest), we can rearrange the equation to solve for time:
time = (final velocity - initial velocity) ÷ acceleration
Since the car is braking, the acceleration is negative, so:
time = (0 - 14.0) ÷ (-a)
where a is the acceleration.
We can use the distance formula:
distance = (initial velocity × time) + (0.5 × acceleration × time²)
Since we know the distance (55.0 m) and the initial velocity (14.0 m/s), we can solve for the acceleration:
55.0 = (14.0 × t) + (0.5 × a × t²)
Using the formula for time we found earlier, we can substitute and solve for a:
55.0 = (14.0 × [(0 - 14.0) ÷ (-a)]) + (0.5 × a × [(0 - 14.0) ÷ (-a)]²)
Simplifying:
55.0 = (14.0 × (14.0 ÷ a)) + (0.5 × a × (14.0 ÷ a)²)
Multiplying through by a²:
55.0a² = (14.0 × a × 14.0) + (0.5 × a × 14.0²)
Simplifying further:
55.0a² = 196a + 98a
55.0a² = 294a
Dividing through by a:
55.0a = 294
a = 5.345 m/s²
Now that we have the acceleration, we can use the first formula to find the retarding force:
retarding force = (mass × change in velocity) ÷ time
The change in velocity is just the initial velocity (14.0 m/s), since the car comes to rest.
The time is the time it took for the car to come to rest, which we found earlier:
time = (0 - 14.0) ÷ (-a) = 2.616 s
Substituting:
retarding force = (760 × 14.0) ÷ 2.616
retarding force = 4051.72 N
Therefore, the retarding force on the car is 4051.72 N.
A baby girl of mass 10kg runs at a speed of 2.0ms-1 . If her acceleration is 0.5ms-1 , calculate (i) the force on her
( ii) the time taken by her to cover a distance of 50m .
(i) The force on the baby girl can be calculated using the formula:
force = mass × acceleration
Substituting the given values:
force = 10 kg × 0.5 ms^-1
force = 5 N
Therefore, the force on the baby girl is 5 N.
(ii) We can use the formula:
distance = (initial velocity × time) + (0.5 × acceleration × time²)
to find the time taken by the baby girl to cover a distance of 50 m.
Since the baby girl is running at a constant acceleration, we can assume the acceleration is the same throughout the 50 m distance.
Substituting the given values:
50 = (2.0 × t) + (0.5 × 0.5 × t²)
Simplifying:
50 = 2t + 0.25t²
Multiplying through by 4:
200 = 8t + t²
Rearranging:
t² + 8t - 200 = 0
We can solve this quadratic equation using the quadratic formula:
t = (-b ± √(b² - 4ac)) ÷ 2a
where a = 1, b = 8, and c = -200
Substituting:
t = (-8 ± √(8² - 4 × 1 × -200)) ÷ 2 × 1
t = (-8 ± √(864)) ÷ 2
Taking only the positive root (since time cannot be negative):
t = (-8 + √864) ÷ 2
t = 7.37 s (rounded to two decimal places)
Therefore, the time taken by the baby girl to cover a distance of 50 m is 7.37 s.
To calculate the retarding force on the car, we can use Newton's second law of motion, which states that force (F) is equal to mass (m) multiplied by acceleration (a):
F = m * a
The initial velocity of the car is 14.0 m/s, and it comes to rest, so the final velocity is 0 m/s. The distance covered by the car is 55.0 m.
Since the car comes to rest, we can use the equations of motion to find the acceleration:
(vf^2) = (vi^2) + 2 * a * d
0^2 = (14.0^2) + 2 * a * 55.0
0 = 196 + 110a
110a = -196
a = -1.78 m/s^2
Now, we can substitute the values of mass (m = 760 kg) and acceleration (a = -1.78 m/s^2) into the equation:
F = m * a
F = 760 kg * (-1.78 m/s^2)
F = -1352.8 N
Therefore, the retarding force on the car is -1352.8 N (opposite to the direction of motion).
To calculate the retarding force on the car, we need to use Newton's second law of motion, which states that the sum of the forces acting on an object is equal to the mass of the object multiplied by its acceleration. In this case, the acceleration of the car is due to the retarding force applied by the brakes.
To determine the acceleration, we can use the equation of motion:
v_f^2 = v_i^2 + 2aΔx
where:
v_f is the final velocity (which is zero in this case, as the car comes to rest),
v_i is the initial velocity (14.0 m/s),
a is the acceleration,
Δx is the distance traveled (55.0 m).
Rearranging the equation to solve for acceleration (a), we have:
a = (v_f^2 - v_i^2) / (2Δx)
Substituting the given values:
a = (0^2 - 14.0^2) / (2 * 55.0)
a = (-196.0) / (110.0)
a = -1.78 m/s^2
Note that the negative sign indicates that the acceleration is in the opposite direction of the initial velocity.
Now, we can calculate the retarding force using Newton's second law:
F = m * a
where:
F is the force (retarding force),
m is the mass of the car (760 kg),
a is the acceleration (-1.78 m/s^2).
Substituting the values:
F = 760 kg * (-1.78 m/s^2)
F = -1352.8 N
The retarding force acting on the car is approximately -1352.8 N. The negative sign indicates that the force is acting in the opposite direction of motion.