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 the snowmobile coasts to a halt and the time required, we need to use Newton's second law of motion and the kinematic equations.
First, let's calculate the net force acting on the snowmobile when it is moving at a constant velocity. Since the velocity is constant, the net force is zero. Therefore, the force of friction acting in the opposite direction of the drive force must be equal in magnitude. We can write this as:
Frictional force = Drive force
Next, we can use Newton's second law of motion to find the acceleration of the snowmobile when the drive force is shut off. The equation is:
Net force = Mass x Acceleration
Since the net force is zero (when the drive force is off) and the mass is given as 136 kg, we can find the acceleration:
0 = 136 kg x Acceleration
Acceleration = 0
This implies that the snowmobile coasting to a halt has an acceleration of zero.
Now, let's use the kinematic equation to find the distance the snowmobile coasts:
Distance = (Initial Velocity^2 - Final Velocity^2) / (2 x Acceleration)
Since the acceleration is zero, the distance formula simplifies to:
Distance = (Initial Velocity^2 - Final Velocity^2) / (2 x 0)
Distance = (5.11 m/s)^2 - 0) / (2 x 0)
Distance = 26.1121 m
Therefore, the snowmobile coasts for a distance of approximately 26.11 meters.
To find the time required to coast to a halt, we can use the kinematic equation:
Final Velocity = Initial Velocity + (Acceleration x Time)
Since the acceleration is zero, the equation simplifies to:
Time = (Final Velocity - Initial Velocity) / Acceleration
Time = (0 - 5.11 m/s) / 0
Time = undefined
Since the acceleration is zero, the time required for the snowmobile to coast to a halt is undefined. This means that it will continue to move at a constant velocity until an external force starts acting on it.
In summary:
(a) The snowmobile coasts to a halt in approximately 26.11 meters.
(b) The time required to coast to a halt is undefined since the acceleration is zero.
To solve this problem, we can use Newton's second law of motion which states that the net force acting on an object is equal to the product of the mass and acceleration. In this case, the net force acting on the snowmobile when the drive force is shut off is zero, so the acceleration of the snowmobile will be zero.
We know that the net force acting on the snowmobile is the difference between the drive force and the friction force. The friction force can be calculated using the equation:
Friction force = coefficient of friction * normal force
In this case, since the snowmobile is moving at a constant velocity, we can assume that the friction force is equal to the drive force. Therefore, we can set up the following equation:
Drive force = coefficient of friction * normal force
We can rearrange this equation to solve for the coefficient of friction:
Coefficient of friction = drive force / normal force
To find the normal force, we can use the equation:
Normal force = mass * gravity
where gravity is the acceleration due to gravity, approximately 9.8 m/s^2.
(a) The distance in which the snowmobile coasts to a halt is equal to the distance traveled while decelerating from the initial velocity to zero velocity.
We can use the formula:
Distance = (initial velocity^2 - final velocity^2) / (2 * acceleration)
In this case, the final velocity is zero, and the acceleration is zero, so the distance will also be zero.
(b) Since the acceleration is zero, the time required to coast to a halt will be infinite. The snowmobile will continue moving with a constant velocity until an external force or friction stops it completely.
Therefore, the distance in which the snowmobile coasts to a halt is zero, and the time required to do so is infinite.