Under the influence of its drive force, a snowmobile is moving at a constant velocity along a horizontal patch of snow. When the drive force is shut off, the snowmobile coasts to a halt. The snowmobile and its rider have a mass of 136 kg. Under the influence of a drive force of 245 N, it is moving at a constant velocity whose magnitude is 5.11 m/s. The drive force is then shut off. Find (a) the distance in which the snowmobile coasts to a halt and (b) the time required to do so.
To find the distance and time required for the snowmobile to coast to a halt, we can use the equations of motion.
First, let's determine the acceleration of the snowmobile while it's under the influence of the drive force. We can use Newton's second law of motion, which states that the acceleration of an object is equal to the net force acting on it divided by its mass:
F = m * a
Where F is the force, m is the mass, and a is the acceleration.
Rearranging the equation, we can solve for acceleration:
a = F / m
Substituting the given values, we have:
a = 245 N / 136 kg
a ≈ 1.80 m/s² (approximately)
Since the snowmobile is initially moving at a constant velocity, the deceleration when the drive force is shut off will be equal to this acceleration, but in the opposite direction.
(a) To find the distance the snowmobile coasts to a halt, we can use the equation of motion:
v² = u² + 2 * a * s
Where v is the final velocity, u is the initial velocity, a is the acceleration, and s is the distance.
Rearranging the equation, we can solve for distance:
s = (v² - u²) / (2 * a)
Substituting the given values, we have:
s = (0 - (5.11 m/s)²) / (2 * -1.80 m/s²)
s ≈ 7.01 m (approximately)
So, the snowmobile coasts to a halt over a distance of approximately 7.01 meters.
(b) To find the time required for the snowmobile to coast to a halt, we can use the equation:
v = u + a * t
Where v is the final velocity, u is the initial velocity, a is the acceleration, and t is the time.
Since the final velocity is zero (as the snowmobile comes to a halt), we get:
0 = (5.11 m/s) + (-1.80 m/s²) * t
Solving for time, we have:
t = -5.11 m/s / -1.80 m/s²
t ≈ 2.84 seconds (approximately)
So, it takes approximately 2.84 seconds for the snowmobile to coast to a halt.
Therefore, the distance in which the snowmobile coasts to a halt is approximately 7.01 meters, and the time required to do so is approximately 2.84 seconds.
To find the distance in which the snowmobile coasts to a halt, we need to use the concept of kinetic friction. The friction force opposes the motion of the snowmobile until it comes to a stop.
(a) Finding the coefficient of kinetic friction:
We can start by finding the coefficient of kinetic friction (μ) using the known driving force and the velocity before shutting off the drive force.
v = 5.11 m/s
F_drive = 245 N
The driving force is balanced by the force of kinetic friction:
F_drive = μ * m * g
Rearranging the equation, we get:
μ = F_drive / (m * g)
Where:
m = 136 kg (mass of the snowmobile and rider)
g = 9.8 m/s^2 (acceleration due to gravity)
Plugging in the values:
μ = 245 N / (136 kg * 9.8 m/s^2)
μ ≈ 0.175
Now, we can calculate the distance the snowmobile coasts to a halt using the formula for distance covered when decelerating uniformly:
d = (v^2) / (2 * a)
where:
d = distance (to be calculated)
v = initial velocity (5.11 m/s)
a = acceleration (deceleration due to friction)
(b) Finding the deceleration:
The deceleration is given by:
a = -μ * g
Substituting the values:
a = -(0.175) * (9.8 m/s^2)
a ≈ -1.715 m/s^2
Now, we can calculate the distance:
d = (5.11 m/s)^2 / (2 * (-1.715 m/s^2))
d ≈ 7.66 m
Therefore, the snowmobile coasts to a halt in approximately 7.66 meters.
To find the time required to coast to a halt, we can use the equation:
t = v / a
where:
t = time
v = initial velocity (5.11 m/s)
a = acceleration (deceleration due to friction)
Plugging in the values:
t = 5.11 m/s / (-1.715 m/s^2)
t ≈ -2.98 s
Note that the negative sign indicates the direction of the acceleration (opposite to the initial velocity) but is not used for time calculations. Therefore, the time required to coast to a halt is approximately 2.98 seconds.
a. a = F/m = 245/136 = 1.80 m/s^2.
d = (V^2-Vo^2)/2a.
d = (0-(5.11)^2)/3.6 = 7.25 m.
b. T = (V-Vo)/a = (0-5.11)/-1.80 = 2.84 s.