A 820-{\rm kg} sports car collides into the rear end of a 2700-{\rm kg} SUV stopped at a red light. The bumpers lock, the brakes are locked, and the two cars skid forward 3.0 {\rm m} before stopping. The police officer, estimating the coefficient of kinetic friction between tires and road to be 0.80, calculates the speed of the sports car at impact.
To calculate the speed of the sports car at impact, we can use the conservation of linear momentum.
The linear momentum before the collision is equal to the linear momentum after the collision, assuming no external forces are acting on the system:
(m1 * v1) + (m2 * v2) = (m1 * u1) + (m2 * u2)
Where:
m1 = mass of the sports car = 820 kg
v1 = initial velocity of the sports car (before collision)
m2 = mass of the SUV = 2700 kg
v2 = initial velocity of the SUV (before collision)
u1 = final velocity of the sports car (after collision)
u2 = final velocity of the SUV (after collision)
Since the SUV is stopped at a red light, its initial velocity (v2) is 0 m/s.
We need to find v1, the initial velocity of the sports car at impact. We also know that the cars skid forward 3.0 m before stopping.
To solve this, we need an additional equation to relate velocities to the distance traveled. We can use the equation for calculating the work done by friction:
Work done by friction = kinetic friction force * distance
The work done by friction can also be calculated as the change in kinetic energy:
Work done by friction = ΔKE
ΔKE = (1/2) * m1 * (v1^2 - u1^2) + (1/2) * m2 * (v2^2 - u2^2)
Also, the kinetic friction force can be calculated as the coefficient of kinetic friction multiplied by the normal force:
Friction force = coefficient of kinetic friction * normal force
In this case, the friction force is equal to the net force acting on the system:
Friction force = (m1 + m2) * g
Where g is the acceleration due to gravity.
Now we can plug this into the equation for work done by friction:
(m1 + m2) * g * d = (1/2) * m1 * (v1^2 - u1^2) + (1/2) * m2 * (v2^2 - u2^2)
Let's calculate the speed of the sports car at impact step by step:
Step 1: Calculate friction force
Friction force = (m1 + m2) * g
= (820 kg + 2700 kg) * 9.8 m/s^2
= 35260 N
Step 2: Calculate work done by friction
(m1 + m2) * g * d = ΔKE
35260 N * 3.0 m = (1/2) * 820 kg * (v1^2 - u1^2) + (1/2) * 2700 kg * (0 - u2^2)
Step 3: Simplify the equation
105780 Nm = (1/2) * 820 kg * (v1^2 - u1^2)
+ (1/2) * 2700 kg * (-u2^2)
Step 4: Since the SUV is stopped, u2 = 0 m/s
105780 Nm = (1/2) * 820 kg * (v1^2 - u1^2)
Step 5: Rearrange the equation to solve for v1
v1^2 = 2 * (105780 Nm) / (820 kg)
v1^2 = 257.439
v1 = √257.439
v1 ≈ 16.04 m/s
Therefore, the speed of the sports car at impact is approximately 16.04 m/s.
To calculate the speed of the sports car at impact, we can start by using the principle of conservation of momentum. According to this principle, the total momentum before the collision should be equal to the total momentum after the collision.
The momentum of an object is given by the product of its mass and velocity. So, we can write the equation for conservation of momentum as:
(m1 * v1) + (m2 * v2) = (m1 * v1') + (m2 * v2')
Where:
m1 = mass of the sports car
v1 = velocity of the sports car before the collision
m2 = mass of the SUV
v2 = velocity of the SUV before the collision
v1' = velocity of the sports car after the collision (which we want to find)
v2' = velocity of the SUV after the collision (which is assumed to be zero since it stopped)
Given values:
m1 = 820 kg
v2 = 0 m/s (since the SUV is stopped)
m2 = 2700 kg
We need to find v1, the velocity of the sports car before the collision. To do this, we can use the equation of motion that relates displacement, initial velocity, final velocity, and coefficient of kinetic friction:
v^2 = u^2 + 2 * a * s
Where:
v = final velocity (which is zero since the cars stopped)
u = initial velocity (which is what we need to find)
a = acceleration (which can be calculated using the coefficient of kinetic friction)
s = displacement (given as 3.0 m)
The frictional force can be calculated by the equation:
f = u * N
Where:
f = frictional force
u = coefficient of kinetic friction (given as 0.80)
N = normal force (which is equal to the weight of the car, given by m * g, where g is the acceleration due to gravity)
Now let's solve for v1, the velocity of the sports car before the collision:
1. Calculate the frictional force:
f = u * N
N = m1 * g
g = 9.8 m/s^2 (standard acceleration due to gravity)
f = u * N
= 0.80 * (820 kg * 9.8 m/s^2)
2. Calculate the acceleration of the car:
The net force acting on the car is given by the frictional force:
F = m1 * a
f = m1 * a
3. Calculate acceleration using the equation:
a = f / m1
4. Use the equation of motion to solve for initial velocity:
v^2 = u^2 + 2 * a * s
Since the final velocity is zero (cars stopped):
0 = u^2 + 2 * a * s
Solve this equation for u, which will give you the initial velocity (v1) of the sports car.
Once you have the initial velocity (v1), you can calculate the speed of the sports car by taking the absolute value of v1.
Note: The negative sign in front of v2 in the earlier equation accounts for the opposite direction of velocities.